Solving a decision problem with graded rewards

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Solving a Decision Problemwith Graded RewardsJose A. Herencia*´Department of Matematicas, Facultad de Ciencias, Univ. de Cordoba, Spain´ ´

M. Teresa Lamata†Department Ciencias de la Computacion e I.A. E.T.S.I. Informatica, Univ. de´ ´Granada, Spain

There are different mathematical models for the concepts of fuzziness, ambiguity,nonspecificity, etc., These models are generally very useful in order to consider realistic

Ž .formulations of decision problems which appear in a wide variety of applications . In thispaper we analyze how to obtain the solution of a decision problem when the rewards are

Žgiven by Zadeh’s graded numbers which can be also applied to rewards given by Zadeh’s.fuzzy numbers . Q 1999 John Wiley & Sons, Inc.

1. INTRODUCTION

Since its foundation in 1965 by Zadeh,1 the theory of fuzzy sets hasexhibited a great growth and has obtained a great importance. This fact is due,fundamentally, to the ability of fuzzy sets to express the ambiguity and vague-ness which are inherent in the human language and thought. Such characteris-tics are present in the processes of decision which appear in multiple real andpractical situations. All this has given rise to the extensive development ofdecision theory in a fuzzy environment.

A very important contribution to the use of fuzzy sets in decision problemsis due to Bellman and Zadeh2; where it is considered how fuzzy sets can be usedto describe the constraint on the alternatives as well as the objectives in certaindecision problems. The corresponding solution is then given by the fuzzy setobtained as the ‘‘confluence’’ of constraints and objectives. These ideas havebeen used as a starting point in many other further researches.

Other procedures have also been developed as additional or alternatives toBellman and Zadeh’s ideas. Thus, fuzziness has been considered in diverseformulations of classical problems of decision, games, optimization, mathemati-

* E-mail: [email protected]† E-mail: [email protected]

Ž .INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 14, 21]44 1999Q 1999 John Wiley & Sons, Inc. CCC 0884-8173r99r010021-24

HERENCIA AND LAMATA22

cal programming, and related theories. Such a fuzziness can be present inŽdifferent aspects such as alternatives, states of Nature, action of the adver-

.saries, constraints, objectives, costs, . . . of the problem considered.Among the different characteristics which appear in those different models,

let us highlight two important facts:

v The extensive use of fuzzy numbers, as an adequate tool to express uncertainquantities and data.

v The representation of the fuzzy problems considered via a-cuts, thus consisting ofa family of crisp problems.3] 5

These two facts have been useful in a number of ways, such as: In thegeneral formulation of the decision problem,6 in fuzzy mathematical program-ming,7 and especially in fuzzy linear programming. In this case, the fuzziness canbe considered in the constraints,8 ] 11 in the costs12,13 and in the technologicalmatrix.14,15 The representation by a-cuts has also been useful in the study ofseveral questions related to the fuzzy linear programming problem as duality,16

integer linear programming17 or multiobjective linear programming.18,19

In this paper, we shall restrict ourselves to the study of a classical problemŽof decision-making with a finite crisp set of states of Nature and with a finite

.crisp set of alternatives , but we assume that the rewards are stated by ‘‘Zadeh’sgraded numbers.’’20,21 These uncertain numbers are analogous to Zadeh’s fuzzynumbers, but their definition and arithmetic is directly based upon the use of

Žthe different level sets as happens with the more general concept of ‘‘gradedset.’’20,22 Therefore, with the use of the graded numbers, we take the two factshighlighted above together, which leads us to study the problem mentionedabove.

In general terms, the advantages of the graded numbers are:

v They are not restricted to the conditions given by Negoita and Ralescu’s repre-sentation theorem23 for the a-cuts of fuzzy numbers.

v 24,25Their operations are defined as a direct generalization of interval analysis.ŽWhile the operations with fuzzy numbers defined via Zadeh’s extension princi-

. 26ple need additional conditions to be done by means of their a-cuts.

In this sense, graded numbers can be manipulated more easily than fuzzynumbers. Nevertheless, there is a strong relationship between graded numbersand fuzzy numbers. All these reasons lead us to propose graded numbers as a

Ž .tool in decision and optimization in a fuzzy environment as we did in Ref. 27 .In this respect, graded numbers can be used in two different ways:

v However, as a concept that can be used directly to model the uncertainty involvedin the decision problem.

This is what we do in this paper, considering that the rewards are gradednumbers.

v However, as an indirect tool that can be used to obtain properties and methodswhen the uncertain data in the problem given are expressed as fuzzy numbers.

SOLVING A DECISION PROBLEM 23

Associated with any fuzzy number, we can consider the graded number formedw xby its weak a-cuts or the graded number formed by its strong a-cuts or any

Žother graded number associated with that fuzzy number which must be an.intermediate one; see Ref. 21 . If we assume that the fuzzy or graded numbers

used must verify certain properties, then we must restrict our choice, discardingsome of those possibilities. In particular, it is usual to impose upper semicontinu-

w xity to the membership functions of the fuzzy numbers. In such a case, the weaka-cuts are closed real intervals, such that they will be the intervals that we shalluse for the definition of graded numbers in this paper. So, in this case, there is anisomorphism between fuzzy numbers and graded numbers. In any other case, theproblem of decision-making given with fuzzy numbers as data can be solved viathe analogous problem where such fuzzy numbers are changed by certain gradednumbers associated with them. Therefore, the methods considered in this paperare directly applicable to a decision problem with fuzzy rewards.

The paper is organized as follows. First, in Section 2, we summarize thedefinition and properties of the graded numbers, considering only those aspectsthat will be useful in the following. In Section 3, we shall consider the orderbetween graded numbers, which is an aspect especially needed to solve ourproblem. Finally, in Section 4, we apply the previous tools and results to solve adecision-making problem with graded rewards.

2. DEFINITION AND PROPERTIES OFZADEH’S GRADED NUMBERS

2.1. Fuzzy Sets and Graded Sets

Let N, R, and I, respectively, denote the set of positive integers, the set ofw x Ž . Ž .real numbers and the unit interval 0, 1 . For any set X, let PP X and FF X ,

respectively, denote the sets of crisp and fuzzy parts of X. Any crisp subsetŽ .S g PP X is identified with its characteristic function x and any fuzzy subsetSŽ .m g FF X is identified with its membership function m: X ª I. The extension of

Ž 2 . Ž . Ž . warbitrary maps f : X ª Y resp., f : X ª Y to f : PP X ª PP Y resp.,Ž .2 Ž .x Ž .PP X ª PP Y is performed in the usual way, while the extension f : FF X ªŽ . w Ž .2 Ž .xFF Y resp., FF X ª FF Y is performed using Zadeh’s extension principle. For

Ž . w a x Ža . w xany a g I and any m g FF X , let m and m , respectively, denote the weaka-cut and the strong a-cut of m.

Ž .DEFINITION 2.1 Ref. 22, Definitions 3.1 and 3.2 . Gi en any set X, we gi e theŽ .term graded subset or part of X or simply graded set to any mapping

Ž . w Ž . Ž .xc : I ª PP X , such that ;a , b g I, a - b « c a = c b . We say that c isŽ . Ž .normal when c a / B, ;a g I. Let GG X denote the set of graded parts of X.Ž . Ž . Ž . Ž . Ž .The inclusion PP X : GG X is gi en by S ¨ S , where S a [ S, ;a g I.I I

Any property or concept ‘‘C’’ is extended from crisp sets to graded sets,wheneër possible, following the criterion,

c is C :m c a is C ;a g IŽ .


w Ž .xIn particular, we define ;c , c , f g GG X :i

v Ž . Ž .c s F c :m c a s F c a , ;a g I.ig I i ig I iv Ž . Ž .c s D c :m c a s D c a , ;a g I.ig I i ig I iv Ž . Ž .c : f:m c a : f a , ;a g I.v Ž . Ž .Any map f : X ª Y is extended to f : GG X ª GG Y defined by

f c a [ f c a ;a g IŽ .Ž . Ž .Ž .

v 2 2Ž . Ž .Any map f : X ª Y is extended to f : GG X ª GG Y defined by

f c , f a [ f c a , f a ;a g IŽ .Ž . Ž . Ž .Ž .

v ŽFor the case X s R, we say that c is convex resp., closed, bounded below, or. Ž . Žbounded above when, ;a g I, c a is conëx resp., closed, bounded below, or.bounded aboë .

Ž . ŽIt is obvious that c is normal iff c 1 / B and that c is bounded below. Ž .andror above iff c 0 also is.

The main result of the relationship between graded sets and fuzzy sets isw w0x Ž1. Ž .xthe following hereafter, we assume that m s X and m s B, ;m g FF X :

Ž .THEOREM 2.2 Ref. 22, Theorem 4.3 . For any set, X, and any map, c : I ªŽ .PP X , we haë:

Ž .1 c is a graded set, if and only if there exists only one fuzzy set, m, ërifying

mŽa . : c a : mwa x ;a g IŽ .

Ž .2 The fuzzy set considered aboë is determined by

m x s sup a g I: x g c a ; x g X� 4Ž . Ž .

Ž .and its a-cuts are ;a g I ,

mwa x s F c b : 0 F b - a� 4Ž .

mŽa . s D c b : a - b F 1� 4Ž .

Ž . Ž .DEFINITION 2.3 Ref. 21, Definition 2.3 . Gi en any c g GG X , we defineŽ . Ž .according to the preïous theorem the fuzzy set m g FF X asc

m x [ sup a g I: x g c a ; x g X� 4Ž . Ž .c

We say that m is the fuzzy set associated with the graded set c . Conërsely,c

Ž .we say that c is a graded set associated with a gi en m g FF X , when m s m.c

The operations and relations given for graded sets are related to thecorresponding ones defined by Zadeh for fuzzy sets. Specifically, in Ref. 22 the


following results are proven:

Ž .PROPOSITION 2.4. For any sets X, Y and any c , c , f g GG X , we haë:i

Ž .1 m s F m .F c ig I ci g I i iŽ .2 m s D m .D c ig I ci g I i iŽ .3 c : f « m : m .c fŽ . Ž . Ž . Ž . Ž .4 The extensions GG X ª GG Y and FF X ª FF Y of any map f : X ª Y ërify

thatm s f mŽ .f Žc . c

Ž . Ž .2 Ž . Ž .2 Ž . 25 The extensions GG X ª GG Y and FF X ª FF Y of any map f : X ª Y ërifythat

m s f m , mŽ .f Žc , f . c f

Ž .COROLLARY 2.5. For any set X and any m, n g FF X , we haë the inclusionm : n , if and only if there are two graded sets, c , associated with m and fassociated with n , such that c : f.

Ž .The concept of a-cut or level cut of a fuzzy set has been useful since thestart of the fuzzy sets theory. Thus, the ‘‘representation method,’’ consisting ofrepresenting each fuzzy problem by a family of crisp problems, via the corre-

Ž .sponding a-cuts see Refs. 3]5 , has become very useful when applied to severaltopics. The concept of graded set is based upon that general methodology,providing some improvements.20 For example, the previous results suggestextending the representation method, using arbitrary graded sets, not necessarilyequal to the family of weak a-cuts.

2.2. Fuzzy Numbers and Graded Numbers

In view of their different applications, there are several definitions for thew Ž .concept of fuzzy number which impose certain restrictions to a m g FF R in

xorder to be considered a fuzzy number . Basically, these definitions can beŽreduced to three as general kinds of fuzzy numbers which allow few modifica-.tions; some of them are considered in Refs. 20 and 21 :

Ž . Ž Ž . Ž .. ŽDEFINITION 2.6. The sets FF R resp, FF R , FF R of Zadeh’s resp., Hutton’sZ H D.Hohle’s fuzzy numbers are defined as¨

FF R [ m g FF R : m is conëx , normal and u.s.c., and mŽ0. is bounded ,� 4Ž . Ž .Z

FF R [ m g FF R : m is nonincreasing, normal and u.s.c.,�Ž . Ž .H

and mŽ0. is bounded aboë ,4FF R [ m g FF R : m is nondecreasing , normal and u.s.c.,�Ž . Ž .D

and mŽ0. is bounded below .4Ž .Hereafter, u.s.c. stands for upper semicontinuous


Ž .Zadeh’s fuzzy numbers defined in Ref. 28 are commonly used in artificialintelligence and they have well suited arithmetic and algebraic properties.Hutton’s fuzzy numbers are used in the so-called fuzzy real line, a veryimportant example of fuzzy topological space. For these fuzzy numbers, wefollow Lowen,29 considering only the left-continuous member in each equiva-

Žlence class thus avoiding the equivalence relation used in Hutton’s original. 30,31definition . We also consider those fuzzy numbers whose membership func-

tion is a distribution function, here called Hohle’s numbers. These fuzzy num-¨bers are used, for example, by Hohle,32 Lowen,33 and Dubois and Prade.34¨

Ž .The operations in FF R are performed via the Zadeh’s extension principle.Z29 Ž .Lowen proved that this can also be done in FF R , for the operations definedH

previously by Rodabaugh,35 in the fuzzy real line.In accordance with these three kinds of fuzzy numbers, the corresponding

three kinds of graded numbers are defined as follows:

DEFINITION 2.7. We consider the following classes of real inter als,

w xCC [ A , B : A , B g R, A F B� 4CC [ y`, A : A g R� 4Žy`

CC [ A , q` : A g R� 4.q`

Ž . Ž Ž . Ž .. Žand we define the sets GG R resp., GG R , GG R of Zadeh’s resp., Hutton’s,Z H D.Hohle’s graded numbers as¨

GG R [ c g GG R : ;a g I, c a g CC� 4Ž . Ž . Ž .Z

GG R [ c g GG R : ;a g I, c a g CC� 4Ž . Ž . Ž .H y`

GG R [ c g GG R : ;a g I, c a g CC� 4Ž . Ž . Ž .D q`

Graded numbers are graded subsets of R. Therefore, we can consideroperations and partial orders between them using Definition 2.1. This enables usto obtain results that are easy to prove. For example, in Refs. 20 and 21, thefollowing aspects are considered:

v Relationships between the three kinds of graded numbers. There is an obviousŽ . Ž .bijection between GG R and GG R , which can be expressed associating to anyD H

ŽHohle’s graded number c the Hutton’s graded number f s yc which is¨.defined via the extension of the mapping R ª R: x ¬ yx . On the other hand,

each Zadeh’s graded number c can be represented by a pair of graded numbers,a Hohle’s one, c , and a Hutton’s one, c , using the following expression,¨ D H

c s c l cD H

Therefore, each Zadeh’s graded number can be represented by a pair of Hutton’sgraded numbers or by a pair of Hohle’s graded numbers.¨


v Usual operations that can be performed with the three kinds of graded numbers.In general, it is necessary to impose certain conditions on a function f : R ª RŽ 2 . Ž . Žresp., f : R ª R in order to obtain, via its extension to GG R , a unitary resp.,

.binary operation in the set of graded numbers used. For Zadeh’s gradednumbers, the continuity of f is suffice. However, for Hutton and Hohle graded¨

Ž .numbers, additional properties of f are required. Therefore, GG R allows forZŽ . Ž .more operations than GG R and GG R . For example, the extensions of theH D

Ž . Ž .product f x, y s xy or the subtraction f x, y s x y y cannot be performedeither with Hutton or with Hohle numbers. Moreover, every usual operation in¨

Ž . Ž . Ž .GG R has a practical sense. This is not the case in either GG R or in GG R . ForZ H DŽ . < < Ž . 2 w Ž . < <example, the extensions of f x s y x or f x s yx resp., f x s x or

Ž . 2 x Ž . w Ž .xf x s x have no practical interest in GG R resp., GG R when applied toH DŽ .graded numbers corresponding to positive resp., negative quantities. This disad-

vantage disappears when the function f is monotonic. In this case, in addition,Ž .the corresponding operation can be performed in GG R by means of theZ

representation of Zadeh’s graded numbers as the intersection of Hutton numberswith Hohle numbers.¨

v Partial order between the three kinds of graded numbers. It is natural to define inŽ .GG R the following partial order,Z

c F f : m c : f and c = f 1Ž .H H D D

This partial order can be extended to the three kinds of graded numbers, beingŽ .compatible in the usual sense with the following operations: product of a real

number with a graded number, sum of graded numbers, minimum and maximumof graded numbers.

Ž .In this paper we are interested in constructing a total order in GG R compati-Zble with the previous partial order. This topic will be considered in Section 3.2.

The analogy between graded numbers and fuzzy numbers is formallyexpressed in the following terms:

Ž . Ž .THEOREM 2.8 Ref. 21, Theorem 4.3 . For any m g FF R and any of the threekinds of numbers corresponding to the cases X s Z, H, D, we haë

m g FF R m 'c g GG R associated with m.Ž . Ž .X X

Therefore, when we work with fuzzy numbers, we have the possibility tochoose graded numbers associated with them, and then work with those gradednumbers. In other terms, the graded numbers can be used to represent problemswith fuzzy numbers, in the same manner as it is usual to consider the families ofweak or strong a-cuts. This general methodology explains the great analogybetween the properties of fuzzy numbers and the properties of graded numbers.

2.3. Zadeh’s Graded Numbers

However, as we have already said in the preceding section, Zadeh’s gradednumbers allow more operations than Hutton numbers or Hohle ones. More¨

Ž .specifically, the former but not the latter numbers verify the following proposi-tion. Their proof stems from the fact that the continuous functions map compactŽ . Ž .resp., connected sets to compact resp., connected sets.


PROPOSITION 2.9.

Ž . Ž . Ž .1 For any continuous function f : R ª R, the extension f : GG R ª GG R proïdes anŽ .inner operation in GG R .Z

Ž . 2 Ž .2 Ž .2 For any continuous function f : R ª R, the extension f : GG R ª GG R proïdesŽ .an inner operation in GG R .Z

Therefore, we can consider the usual unitary and binary operations inŽ . ŽGG R and, by the associativity, the binary operations can be applied to anyZ

. Ž .finite quantity of numbers . The same happens with FF R , which allows moreZŽ . Ž . 36operations than FF R or FF R . Moreover, Delgado, Verdegay, and VilaH D

consider some aspects relating to the semantical interpretation and the practicalapplicability of fuzzy numbers; concluding that Zadeh’s fuzzy numbers are the

Žmost adequate for the applications while Hutton fuzzy numbers offer a moretheoretical and formal representation of the concept of fuzzy number, adequate

.from a topological point of view .In accordance with these facts, from now on, we shall consider only Zadeh’s

numbers. We shall work directly with Zadeh’s graded numbers; which can beapplied to the case of handling Zadeh’s fuzzy numbers, as we said in Section 2.2.

In order to fix the notation used in the following, let us reformulate theŽ .definition of Zadeh’s graded numbers as well as their usual operations and let

us consider some particular cases.�w x 4A Zadeh’s graded number is a map c : I ª A, B : A, B g R , assigning

w Ž . Ž .x wto any a g I the interval a a , b a and verifying that ;a , b g I, a - b «Ž . Ž . Ž . Ž .xa a F a b F b b F b a . Therefore, any Zadeh’s graded number is deter-

mined by a pair of functions a, b: I ª R, verifying the following three condi-tions:

Ž . Ž .1 The function a a is nondecreasing.Ž . Ž .2 The function b a is nonincreasing.Ž . Ž . Ž .3 a 1 F b 1 .

Ž .Obviously, GG R extends R because we can identify any real number PZŽ� 4. Ž .with Zadeh’s graded number P using the notation given in Definition 2.1 ,I

Ž . Ž .determined by the constant functions a a s b a s P, ;a g I.The operations between Zadeh’s graded numbers are obtained from the

Ž .general definition of mapping extension to graded sets Definition 2.1 . In orderŽto solve a decision-making problem via the classical methods as we do in

. Ž .Section 4 , we only need to extend to GG R the following continuous functions,

f : R ª R: x ¬ Px with P g RŽ .f : R2 ª R: x , y ¬ x q yŽ .

2 � 4f : R ª R: x , y ¬ min x , yŽ .2 � 4f : R ª R: x , y ¬ max x , yŽ .


Such extensions are carried out independently for the different levels a g I.Ž .In each level, the extension to crisp subsets here real intervals corresponds to

the usual operations used in the interval analysis. Thus we have, respectively,w Ž .the following usual operations for Zadeh’s graded numbers c a s

w Ž . Ž .x Ž . w Ž . Ž .x Ž . w Ž . Ž .xa a , b a , c a s a a , b a , c a s a a , b a and the real num-1 1 1 2 2 2xber P ,

Pc a s Px : x g c a� 4Ž . Ž . Ž .

Pa a , Pb a if P G 0Ž . Ž .s ½ Pb a , Pa a if P - 0Ž . Ž .

c q c a s x q y : x g c a , y g c a� 4Ž . Ž . Ž . Ž .1 2 1 2

s a a q a a , b a q b aŽ . Ž . Ž . Ž .1 2 1 2

� 4 � 4min c , c a s min x , y : x g c a , y g c a� 4Ž . Ž . Ž .1 2 1 2

s min a a , a a , min b a , b a� 4 � 4Ž . Ž . Ž . Ž .1 2 1 2

� 4 � 4max c , c a s max x , y : x g c a , y g c a� 4Ž . Ž . Ž .1 2 1 2

s max a a , a a , max b a , b a� 4 � 4Ž . Ž . Ž . Ž .1 2 1 2

To conclude this section, let us consider some particular cases of Zadeh’sgraded numbers which may be useful in practical applications.

DEFINITION 2.10. We gi e the terms:

v Triangular graded number to Zadeh’s graded number determined by two linearfunctions a, b: I ª R, whose graphs describe a triangle. More precisely, for any realnumbers A F M F B, we haë the triangular graded number t determined byŽ A , M , B .the functions,

a a s A q a M y A b a s B y a B y MŽ . Ž . Ž . Ž .v 3-graded number to Zadeh’s graded number determined by two step functions

1 1 2 2w x Ž x Ž xa, b: I ª R, which are constant on the inter als 0, , , , and , 1 . Thus, for3 3 3 3any real numbers A F A9 F A0 F B0 F B9 F B, we haë the 3-graded numberj determined by the functions,Ž A , A9, A0 , B 0 , B 9, B .

1a a s A b a s B if 0 F a FŽ . Ž . 3

1 2a a s A9 b a s B9 if - a FŽ . Ž . 3 3

2a a s A0 b a s B0 if - a F 1Ž . Ž . 3

These definitions correspond to simple formulations which can express theimprecise quantities considered in practical applications:

v Triangular graded numbers are analogous to the triangular fuzzy numbers; whichare very frequently used as membership functions associated with linguistic labels.


v The 3-graded numbers can model the situations where an expert gives threeŽintervals as values for an inexact number according to three levels of error, three

.degrees of accuracy, three assumptions of risk, . . . ; in the same manner as threeŽ .sets determine a 3-vague set or 3-flou set .

Nevertheless, for more theoretical aims, we could consider more generalkinds of Zadeh’s graded numbers. For example, we could define, respectively:

v The L-R-graded numbers, fixing some functions L and R as a basis for obtainingŽ . Ž . wthe functions a a and b a , respectively, for the different values of the four

Ž . Ž . Ž . Ž .xparameters a 0 , a 1 , b 1 , and b 0 . The triangular graded numbers would resultŽ . Ž . Ž .as the particular case where L and R are linear, a 0 s A, a 1 s b 1 s M, and

Ž .b 0 s B. Obviously, this kind of graded numbers would be analogous to theknown L-R-fuzzy numbers defined by Dubois and Prade.37

v The n-graded numbers, fixing n values in I: 0 - a - a - ??? - a s 1 and1 2 ntaking a and b as step functions which are constant in the corresponding nsubintervals. This procedure is analogous to the use of the lattice L sŽ .a , a , . . . , a , 1 , instead of the unit interval, taking into consideration the1 2 ny1concept of an L-fuzzy set defined by Goguen.38

3. RANKING GRADED NUMBERS

3.1. Ranking Fuzzy Numbers

Fuzzy numbers, due to their intrinsic vagueness, are not endowed with anatural total order, as is the case for the real numbers. However, in theapplications a certain criterion is usually needed to order the fuzzy numbers.

ŽThus, we can find in the literature several methods to rank fuzzy numbers Refs..39]46 . . . . These methods can be grouped into two categories. On the one

hand, we have the procedures which define the order between fuzzy numbers asa fuzzy relation. On the other hand, we have those which consider a crisp orderrelation between fuzzy numbers.

With the aim of obtaining a crisp order between the alternatives of aŽ .decision problem which enables us to choose the best , we are interested here

in the second kind of methods. Among them, we consider specifically a methoddue to Gonzalez,47,45,48 which is analogous to the method that we use for´Zadeh’s graded numbers. With respect to the levels a , Gonzalez considers a´subset Y : I and an additive measure S on Y. This enables us to assign a certaindegree of importance to the different levels a , which must be fixed in each

Žspecific problem. Then, for any Zadeh’s fuzzy number m with normal and u.s.c..membership function , Gonzalez considers the value,´

lV m [ lB q 1 y l A dS aŽ . Ž . Ž .HS a aY


w x w xwhere A , B is the weak a-cut of m and the parameter l g 0, 1 states thea a

Ž .degree of optimism as we shall see below .When the set Y, the measure S, and the parameter l are chosen, two fuzzy

lŽ . lŽ .numbers m and n are considered indifferent if V m s V n , while m isS SlŽ . lŽ .considered greater than n if V m ) V n .S S

lŽ .Let us point out that the value V m generalizes some other indicesSŽw x.considered in the literature. Thus, if we take Y s I, S A, B s B y A, ;A F

ŽB g R the usual Lebesgue measure, which assigns the same importance to the. Ždifferent levels a and l s 0.5 the equilibrium point between optimism and

. lŽ . 46 Žw x.pessimism , then V m coincides with Yager’s index. Taking Y s I, S A, BSs B2 y A2, ;A F B g R, and l s 0.5, we obtain the Tsumua, Terano, andSugeno index.49

3.2. Partial and Total Orders Between Zadeh’s Graded Numbers

Zadeh’s graded numbers offer a direct generalization of the concept ofcompact real interval. Thus, the arithmetic with graded numbers is a directgeneralization of the arithmetic used in the interval analysis. In the same way,the problem of ranking Zadeh’s graded numbers is similar to the problem of

w xranking intervals. For this reason, based upon an order between intervals A, BŽ .with A F B g R we consider, via the integration on I, an order between

Žgraded numbers finally analogous to the order considered by Gonzalez, defined´via the integration of a function which is applied to the a-cuts of the fuzzy

.numbers .We must extend the usual order of the real line R. Therefore, it is fully

w x w xreasonable to take A , B F A , B when A F A and B F B . Neverthe-1 1 2 2 1 2 1 2less, the situation is not so obvious when one interval is included in the otherone. Let us begin considering only the former situation, since the intervals in thelatter situation are not comparable. In this manner we begin by considering apartial order. After studying its properties, we shall extend this partial order to atotal order.

Ž .With respect to Zadeh’s graded numbers, the partial order defined in 1reflects the case where, ;a g I, the corresponding intervals are comparable andin the same order. Let us write this same definition again, with the terminologyused in the preceding section:

Ž .DEFINITION 3.1. In GG R , we define the following partial order: For any c , cZ 1 2Ž . Ž . w Ž . Ž .x Ž .g GG R , gi en by c a s a a , b a i s 1, 2 we say that c F c whenZ i i i 1 2

;a g I a a F a a and b a F b a 2Ž . Ž . Ž . Ž . Ž .1 2 1 2

Considering the intervals corresponding to each level a , it may immediatelybe checked that this definition extends the usual order in R and that it is


compatible with the usual operations considered earlier:

Ž .PROPOSITION 3.2. For any c , f, j g GG R and for any P, Q g R, the followingZproperties hold:

Ž . Ž� 4. Ž� 4.1 If c s P and f s Q then we haë: c F f m P F Q.I IŽ . Ž .2 ;P g 0, q` , we haë: c F f m Pc F Pf.Ž . Ž .3 ;P g y`, 0 , we haë: c F f m Pf F Pc .Ž . � 4 � 44 c F f m min c , f s c m max c , f s f.Ž .5 c F f « c q j F f q j .

These properties make the partial order F interesting. Nevertheless, if weuse it in a decision-making problem, then we find a drawback: there are manygraded numbers which cannot be compared with respect to this partial order.

ŽWith the aim of extending it to a total order preserving, as much as possible,. �w x 4the preceding properties , we turn again to CC s A, B : A F B g R as a basis

Ž .for the study of GG R .Zw x w xGiven any two intervals A , B and A , B in CC, if they are not1 1 2 2

Žcomparable with respect to the partial order F that is, if it is not true that.A F A and B F B and not vice versa , then one interval must be strictly1 2 1 2

w x w xincluded in the other interval. Let us assume that A , B ; A , B being1 1 2 2A - A F B - B . In these conditions, what interval must be considered the2 1 1 2greatest? In other terms; if those two intervals represent two possible rewards,which is the best? An optimistic individual will consider only the highest values

w xB and B in each interval, thus preferring the interval A , B . On the1 2 2 2contrary, a pessimistic individual will consider only the left extremes A and A ,1 2

w xthus preferring the interval A , B . A half-way point of view will tend to1 1Ž . Ž .compare the mean values A q B r2 and A q B r2. These three alterna-1 1 2 2

tives, as well as the intermediate ones, can be considered together introducing aparameter l g I as a degree of optimism. For each value of l, the intervalw x Ž .A, B will be viewed as the number lB q 1 y l A. Indeed, this number

Ž .coincides with B for the optimistic individual l s 1 , it coincides with A forŽ . Ž .the pessimistic individual l s 0 , and with the mean value A q B r2 for the

1Ž .middle point between optimism and pessimism l s . These facts lead to the2

following:

DEFINITION 3.3. For each l g I, we haë the order F defined in CC as follows:l

w x w x w x w x Ž .A , B F A , B : m A , B F A , B or lB q 1 y l A - lB q1 1 l 2 2 1 1 2 2 1 1 2Ž . Ž w x w x1 y l A thus extending the partial order A , B F A , B : m A F A and2 1 1 2 2 1 2

.B F B .1 2

w x w x ŽLet us note that, if A , B F A , B then, ;l g I, we have that 1 y1 1 2 2. Ž . Ž .l A F 1 y l A and lB F lB resulting that lB q 1 y l A F lB q1 2 1 2 1 1 2

Ž .1 y l A . This allows us to easily check the reflexive, antisymmetric, and2transitive properties of F , since it is really an order. Moreover, for l s 0 andl


Ž .l s 1, it is a total order. Indeed, for l s 0, the case A - A resp., A - A1 2 2 1w x w x Ž w x w x.gives rise to A , B F A , B resp., A , B F A , B , while the case1 1 0 2 2 2 2 0 1 1

ŽA s A is the source of the comparison throughout the order F depending1 2.only on the values B and B . Analogously, any two intervals can be compared1 2

for l s 1. Let us note that F and F are lexicographic orders.0 1Ž .Nevertheless, for any l g 0, 1 , the relation F is not a total order in CCl

w x w xbecause any two different intervals A , B and A , B are still not compared1 1 2 2Ž . Ž .such that lB q 1 y l A s lB q 1 y l A and one is strictly inside the1 1 2 2

Ž .other being A - A F B - B or A - A F B - B . Given any interval2 1 1 2 1 2 2 1w x Ž . w xA , B , this situation happens exclusively for any other interval A , B such1 1 2 2that the differences between their extremes verify the following proportion:

Ž . Ž .B y B s 1 y l rl A y A . However, these differences must have the same2 1 1 2sign and never are zero, being the amplitude of one interval strictly greater thanthe amplitude of the other interval. This fact enables us to compare them insome sense. We think that, in addition to the degree of optimism of an

Ž .individual which is already fixed with the value of l , the characteristic thatgives preference to one of those intervals is that of liking risk. Thus, a riskyindividual will prefer the broader interval; while a safe individual will prefer thenarrower interval, whose values have a higher degree of certainty of beingobtained. Accordingly, in addition to the total orders F and F , we can0 1

Ž . Ždefine ;l g 0, 1 the total orders F and F associated with a risky and alr l s.safe individual, respectively as follows:

Ž .DEFINITION 3.4. For each l g 0, 1 , we haë the total orders F and Flr l sdefined in CC by:

w x w x w x w x Ž . Ž .A , B F A , B : m A , B F A , B or lB q 1 y l A - lB q 1 y l A or1 1 l r 2 2 1 1 2 2 1 1 2 2w Ž . Ž . xlB q 1 y l A s lB q 1 y l A and B y A ) B y A .1 1 2 2 2 2 1 1

w x w x w x w x Ž . Ž .A , B F A , B : m A , B F A , B or lB q 1 y l A - lB q 1 y l A or1 1 l s 2 2 1 1 2 2 1 1 2 2w Ž . Ž . xlB q 1 y l A s lB q 1 y l A and B y A - B y A .1 1 2 2 2 2 1 1

CC is also totally ordered by the orders F and F gi en in Definition 3.3,0 1which can be determined as follows,

w x w x w xA , B F A , B m A - A or A s A and B F B1 1 0 2 2 1 2 1 2 1 2

w x w x w xA , B F A , B m B - B or B s B and A F A1 1 1 2 2 1 2 1 2 1 2

The method that we have followed above establishes that the differentorders defined in CC are successive extensions of F . Thus, it is obvious that, ;I ,1I g CC, the following implications hold:2

v w xI F I « ;l g I, I F I .1 2 1 l 2v Ž . w x;l g 0, 1 , I F I « I F I and I F I .1 l 2 1 l r 2 1 l s 2


Moreover, by definition, we also have the converse of the last implication.Thus, the order F can be obtained as the intersection of both extensions, Fl l rand F . Similarly, the partial order F can be obtained from their extensionslsin the following sense:

PROPOSITION 3.5. For any I , I g CC, the following conditions are equi alent:1 2

Ž .a I F I .1 2Ž .b I F I and I F I .1 0 2 1 1 2Ž .c ;l g I, I F I .1 l 2Ž . Ž .d ;l g 0, 1 , I F I .1 l 2Ž . Ž .e ;l g 0, 1 , I F I .1 l r 2Ž . Ž .f ;l g 0, 1 , I F I .1 l s 2

Ž . Ž .Proof. It is obvious as we said earlier that condition a implies anyone of theŽ . Ž . Ž . Ž .remaining conditions b ] f . It is also immediately to be found that b « a .

w x lFor the remaining cases, let us take I s A , B and let us denote M s lB qi i i i iŽ . Ž .1 y l A i s 1, 2 .i

Ž . Ž .Now, let us prove the sufficiency of c and d :

Ž .c « a : If ;l g I, I F I , then we can consider the particular cases l s 0 and1 l 2Ž . Ž .l s 1, thus fulfilling b which implies a .

Ž . Ž . Ž . l ld « a : Let us assume that ;l g 0, 1 , I F I . Then, ;l g 0, 1 , M F M .1 l 2 1 2Taking into consideration that A - M l - B , lim M l s A andi i i lª 0 i i

l Ž .lim M s B , i s 1, 2 it suffices to take the limits l ª 0 and l ª 1 tolª 1 i iobtain that A F A and B F B as required.1 2 1 2

Ž . Ž .The sufficiency of e and f can be proved with the same argument usedŽ .for d . B

Taking into consideration Definitions 3.3 and 3.4, we now try to obtainŽ .similar orders in GG R . Following Gonzalez, we consider a subset Y : I and an´Z

additive measure S on Y, thus assigning a certain degree of importance to thedifferent levels a , according to the characteristics of each specific problem.

Ž .Then, we summarize via integration on Y by a real compact interval theinformation given in any Zadeh’s graded number. Thus, the Zadeh’s graded

w Ž . Ž .xnumber c is approximated by the interval AA c , BB c defined as follows:S S

DEFINITION 3.6. With respect to any subset Y : I and any additi e measure S onY, we haë the mapping,

GG R ª CC : c ¬ AA c , BB cŽ . Ž . Ž .Z S S

where

AA c [ a a dS a BB c [ b a dS aŽ . Ž . Ž . Ž . Ž . Ž .H HS SY Y


Ž .This mapping induces in GG R the following equi alence relation,Z

c ; f : m AA c s AA f and BB c s BB fŽ . Ž . Ž . Ž .S S S S

Via this mapping, the orders defined in CC induce respective orders in theŽ .quotient set GG R r; . Therefore we have:Z

w xPROPOSITION 3.7. Let us denote by c the equi alence class of any Zadeh’sŽ . Ž .graded number c in the quotient set GG R r; and let us consider ;l g 0, 1 , theZ

lŽ . Ž . Ž . Ž .älue MM c [ l BB c q 1 y l AA c .S S SWith respect to any subset Y : I and any additi e measure S on Y, we haë in

Ž .the quotient set GG R r; the following partial orders:Z

v w x w x Ž . Ž . Ž . Ž .c F f : m AA c F AA f and BB c F BB f .S S S Sv l lŽ . w x w x w x w x Ž . Ž .For each l g 0, 1 , we haë: c F f : m c F f or MM c - MM f .l S S

We also haë the following total orders,

v w x w x Ž . Ž . w Ž . Ž . Ž . Ž .xc F f : m AA c - AA f or AA c s AA f and BB c F BB f .0 S S S S S Sv w x w x Ž . Ž . w Ž . Ž . Ž . Ž .xc F f : m BB c - BB f or BB c s BB f and AA c F AA f .1 S S S S S Sv l lŽ . w x w x w x w x Ž . Ž .For each l g 0, 1 , we haë: c F f : m c F f or MM c - MM f orlr S S

w lŽ . lŽ . Ž . Ž . Ž . Ž .xMM c s MM f and BB f y AA f ) BB c y AA c .S S S S S Sv l lŽ . w x w x w x w x Ž . Ž .For each l g 0, 1 , we haë: c F f : m c F f or MM c - MM f orls S S

w lŽ . lŽ . Ž . Ž . Ž . Ž .xMM c s MM f and BB f y AA f - BB c y AA c .S S S S S S

Ž . Ž .Each one of the previous partial or total orders defined in GG R r; canZŽ .be considered as a partial order in GG R if we consider that any two distinct butZ

Žequivalent Zadeh’s graded numbers c / f, c ; f are not comparable but. Ž .indifferent . Nevertheless we are interested in obtaining a total order in GG R .Z

Ž .To do so, we can extend the order which results in GG R from any total orderZŽ . Žin GG R r; establishing a certain total order between the equivalent or up toZ

.now ‘‘indifferent’’ graded numbers. The rationale behind the proposal of theŽ .‘‘safe’’ and ‘‘risky’’ orders F and F defined in CC can be also used here inls l r

the following sense:Let us consider any two distinct but equivalent Zadeh’s graded numbers c

Ž . Ž . Ž . Ž .and f ; then AA c s AA f and BB c s BB f but c / f. From the latterS S S S� Ž . Ž .4inequality it follows that the set D [ a g I: c a / f a is nonempty. Any

a g D allows us to distinguish c from f and then to order them according toŽ . Ž .the corresponding order between the intervals c a and f a . Among these

possible values of a , a safe individual will consider the greatest one, thusregarding the narrowest intervals. On the contrary, a risky individual willconsider the lowest value of a , thus regarding the broadest intervals, whichpresent a greater degree of risk.

Ž .If we try to use the preceding criterion for all GG R , then we find theZfollowing problem: the set D is not necessarily closed; so, the values max D and


Žmin D may not exist moreover, other values may not exist which can be.considered as reasonable substitutes for max D and min D . For example, let

w xus consider, with respect to the set Y s I and Lebesgue’s measure S A, B s< < Ž .B y A , the triangular graded number t see Definition 2.10 and theŽ0, 1, 2.Zadeh’s graded number c defined by the following functions,

0 a s 0¡1 1 1

- a F n g N2n q 1 2n q 2 2n~a a [Ž . 1 1 11 y 1 y - a F 1 y n g N

2n q 1 2n 2n q 2¢1 a s 1

b a [ 2 y a a ;a g IŽ . Ž .

These two graded numbers coincide for the values of a belonging to the set� 4 � Ž . 4 � Ž . 4J s 0, 1 j 1r 2n q 1 : n g N j 1 y 1r 2n q 1 : n g N , and they are dif-

Ž .ferent in the complementary set D s I _ J. Moreover, the step functions a aŽ .and b a cross over the linear functions which define t in such a way that:Ž0, 1, 2.

1 3v Ž . Ž . Ž . Ž .AA c s AA t s and BB c s BB t s .S S Ž0, 1, 2. S S Ž0, 1, 2.2 2v min D, max D any other value a g D do not exist which can be reasonably

chosen as ‘‘the value used to discriminate’’ between the graded numbers c andt .Ž0, 1, 2.

The preceding situation must be considered as a theoretical counterexam-ple. Obviously, such a situation will not appear in practical applications. Indeed,in practice we shall only need to use Zadeh’s graded numbers determined by a

Žfinite number of parameters such as the triangular graded numbers given bythree parameters, the L-R-graded numbers given by four parameters, or the

.n-graded numbers by 2n ones . In such a case, it will always be possible todetermine ‘‘the greatest’’ and ‘‘the lowest’’ value in D which can be used todiscriminate between two ‘‘indifferent’’ graded numbers, thus ordering them.This is the only situation considered here, because this will be the case in apractical decision-making problem. For such a situation, we consider in thefollowing section the procedure for totally ranking the graded numbers.

3.3. Ranking Zadeh’s Graded Numbers in Practice

Taking into consideration the successive extensions of the partial order Fand the criteria used in the preceding section, let us now establish the resultingmethod to totally order the kind of Zadeh’s graded numbers which will be usedin a decision-making problem.

More specifically, we restrict ourselves to considering those Zadeh’s gradednumbers which can be determined by a finite number of parameters. Let us

wŽ .denote by GG R the set of such ‘‘finite representable’’ Zadeh’s graded num-Z


wŽ .bers. Before obtaining a total order in GG R , we must choose the following:Z

v A subset Y : I and an additive measure S on Y, thus fixing the importanceŽ .assigned to the levels a g I. According to these values we have defined: AA c [S

Ž . Ž . Ž . Ž . Ž . w Ž . w Ž . Ž .xxH a a dS a and BB c [ H b a dS a where c a s a a , b a .Y S Yv lŽ .A degree of optimism l g I. Associated with l we have defined MM c [S

Ž . Ž . Ž .l BB c q 1 y l AA c .S Sv � 4 Ž .An element x g r, s which determines the liking for risk x s r or the liking for

Ž .safety x s s . The value of x acts independently from the degree of optimism land the choice of both parameters can be also considered as independent.Nevertheless, we think that the smaller the degree of optimism, the greater willbe the liking for risk.

wŽ .We assume that the graded numbers belonging to GG R , the subset Y andZthe measure S chosen above, enable us to consider a level a g Y defined as thexlowest, for x s r, and the highest, for x s s, among the levels a g Y such thatŽ . Ž . Ž . Ž . Ž . Ž .c a / f a when AA c s AA f and BB c s BB f .S S S S

Ž .When Y, S, l, and x and so also a are chosen, we can apply the criteriaxdefined in the preceding section to rank Zadeh’s graded numbers. That gives

wŽ .rise to the desired definition of total order in GG R :Z

DEFINITION 3.8. For any additi e measure S on Y : I, any l g I, and any� 4 wŽ .x g r, s , we haë the total order F defined in GG R , as follows:Sl x Z

v w Ž . Ž .x w Ž . Ž .xIf the inter als AA c , BB c and AA f , BB f are different, then they are usedS S S S� 4to rank c and f according to the total order F if l g 0, 1 or the total order Fl l x

Ž . Ž .if l g 0, 1 . See Definitions 3.3 and 3.4 .v w Ž . Ž .x w Ž . Ž .x Ž . Ž .If AA c , BB c s AA f , BB f , then we use the inter als c a and f aS S S S x x

Ž .to rank c and f according to the total orders used in the preceding case .

This kind of total order can be used to rank Zadeh’s graded numbers whichare present in practical applications. It is not a strict or severe criterion. On thecontrary, it covers a variety of criteria, corresponding to the different values ofS, l, and x, thus resulting to be appropriate for the different subjective opinionsof the individual who must rank the numbers given. Nevertheless, for any of thevalues of S, l, and x, the order F extends the partial order F consideredSl x

Ž . Ž .in 1 and 2 as the most natural order between Zadeh’s graded numbers.w Ž . w Ž . Ž .x x Ž .Indeed, if c F c where c a s a a , b a , i s 1, 2 then AA c s1 2 i i i S 1

Ž . Ž . Ž . Ž . Ž . Ž . Ž .H a a dS a F H a a dS a s AA c . Analogously, BB c F BB c .Y 1 Y 2 S 2 S 1 S 2� 4Therefore, c F c , ;l g I, ; x g r, s .1 Sl x 2

� 4Remark. If we consider the extreme case where the singletons Y [ a area

the subsets chosen in I, together with the measure S degenerated on Y , thena a

the order F coincides with the corresponding order defined for the inter-S l xa

Ž .vals c a g CC. Using those extreme situations, we can obtain the partial orderŽ .F defined in GG R as the intersection of the total orders F , in the sameZ Sl x

way as we did in Proposition 3.5 for the orders defined in CC. Nevertheless, for


more natural values of Y and S, we cannot find such a result. For example, letw xY s I, S A, B s B y A, c s t , and f s t . In this case we have:Ž0, 6, 6. Ž5, 5, 11.

w Ž . Ž .x w x w Ž . Ž .x w xAA c , BB c s 3, 6 and AA f , BB f s 5, 8 . Therefore, c F f, ;l gS S S S Sl x� 4I, ; x g r, s , but c g f.

Now, let us consider the compatibility of the order F with the usualSl xorder of R and with the operations sum and product used with graded numbers.In this respect, in addition to the definition of these operations, we must takeinto consideration the following facts. First, the linearity of the integration gives

Ž . Ž . Ž . Ž . Ž .rise to: AA Pc s P AA c , AA c q f s AA c q AA f as well as the analogousS S S S Sequations for BB and for MM l. Second, the product from a positive real numberS SP preserves the order between the extremes and the amplitudes of any twocompact intervals. Third, the product from a negative real number P inverts theorder between the extremes of any two compact intervals, but preservesthe order between their amplitudes. Thus, we have the compatibility of the

Žorder F with the operations sum and product of graded numbers similar toSl x.what happens with the partial order F , as expressed in the Proposition 3.2 ;

except the inversion of the order between two graded numbers c and f, viatheir product by a negative number P, in the only case where such numbers

w Ž . Ž .xmust be ordered according to the amplitude of the intervals AA c , BB c andS Sw Ž . Ž .x Ž . Ž .AA f , BB f or the amplitude of the intervals c a and f a . The rest ofS S x xthe properties are maintained:

� 4PROPOSITION 3.9. For any additi e measure S on Y : I, any l g I, any x g r, s ,wŽ .any c , f, j g GG R and for any P, Q g R, the following properties hold:Z

Ž . Ž� 4. Ž� 4.1 If c s P and f s Q then we haë: c F f m P F Q.I I Sl xŽ . Ž .2 ;P g 0, q` , we haë: c F f m Pc F Pf.Sl x Sl xŽ .3 c F f « c q j F f q j .Sl x Sl x

With respect to the properties considered in Proposition 3.2, in addition tothe exception considered above, let us notice the absence of the compatibilitywith the min and max operations. Indeed, the definition of these operations,

Ž .made ‘‘level by level’’ see Section 2.3 is in accordance with the partial order FŽalso consisting of the uniform ranking of the intervals corresponding to each

.level a ; but such a definition does not cover total order F , which is basedSl xupon the integration on Y and the selection of only one level a . For example,x

w xtaking Y s I and S A, B s B y A, we find that t F t , ;l g I,Ž0, 6, 6. Sl x Ž5, 5, 11.� 4 Ž . � 4; x g r, s , as stated in the preceding remark , while neither min t , tŽ0, 6, 6. Ž5, 5, 11.

� 4nor max t , t are triangular numbers, thus not coinciding neither withŽ0, 6, 6. Ž5, 5, 11.t nor with t .Ž0, 6, 6. Ž5, 5, 11.

wŽ .Given a finite number of elements of GG R , it their minimum and theirZmaximum must exist according to the total order F . In general, as happensSl xin the example considered above, these minimum and maximum will notcoincide with the min and max operations used up until now. We shall denote

Ž . Ž .min resp., max the minimum resp., the maximum associated with theSl x Sl xtotal order F .Sl x


4. THE SOLUTION OF A DECISION PROBLEMWITH GRADED REWARDS

4.1. Formulating and Solving the Problem

However, as we said in the Introduction, the aim of this paper is to solve aclassical decision-making problem where the rewards are stated by Zadeh’sgraded numbers. This section is devoted to that aim, with the help of thedefinitions and results stated in Sections 2 and 3. First, we consider a generalformulation of the problem. Later, we illustrate the method proposed with twoexamples, using the two particular cases of Zadeh’s graded numbers consideredabove: triangular graded numbers and 3-graded numbers.

The formulation of the problem is as follows. We must choose an alterna-tive, in a practical situation where we assume that the following is given:

v � 4A finite crisp set of possible alternatives V s v , v , . . . , v .1 2 Nv � 4A finite crisp set of states of Nature S s s , s , . . . , s , together with a certain1 2 K

body of evidence on such states. In extreme cases, we have ‘‘certainty’’ about SŽ . Žwhen only one element s may appear and ‘‘total uncertainty’’ when all thec

.elements s , k s 1, . . . , K are possible, but none of them appear necessarily . Wekalso consider the ‘‘risk environment,’’ where a probability distribution on S isknown.

v wŽ . Ž .A mapping V = S ª FF R , which assigns to any pair v , s the reward cZ n k nkŽ .for all n s 1, 2, . . . , N, and for all k s 1, 2, . . . , K .

In order to solve this problem, in accordance with the most usual classicalmethods, we follow two steps:

Ž . Ž .1 Associated with each alternative v n s 1, . . . , N we calculate the expectednŽ .reward c , from the vector of possible rewards c , c , . . . , c , based uponn n1 n2 n K

the evidence on S and the subjective degree of optimism.Ž . Ž2 According to the subjective importance given to the different levels a g I fixed

.by the additive measure S on Y : I , the degree of optimism l g I and theŽ � 4.liking for risk fixed by the element x g r, s , we consider the corresponding

wŽ .total order F in the set of graded numbers FF R . Therefore, we chooseSl x Zthe best alternative as the alternative associated with the maximum expectedreward, where such a maximum is determined by this total order. That is, thebest alternative v corresponds to the suffix ‘‘best’’ given bybest

� 4c [ max c , c , . . . , cbest 1 2 NSl x

The parameters S, l, and x chosen for the ranking made in the secondstep, may also be used in the first step for the calculation of the expected

Ž .reward. More specifically, let us determine the value of c n s 1, . . . , Nncorresponding to the three kinds of environment considered above:

Uncertainty. Following Hurwicz’s criterion, we have

� 4 � 4c [ l max c , c , . . . , c q 1 y l min c , c , . . . , cŽ .n n1 n2 n K n1 n2 n K

If l s 0, then we have Wald’s or maximin criterion, being c [n� 4min c , c , . . . , c . On the contrary, if l s 1, then we have the optimistn1 n2 n K

� 4maximax criterion where c [ max c , c , . . . , c .n n1 n2 n K


Remark. Instead of the min and max operations considered above, we can usethe minimum min and the maximum max determined by the order F .Sl x Sl x Sl x

ŽRisk. Let P be the probability associated with the state s for any k s 1, . . . , K thusk k.P G 0 and P q P q ??? qP s 1 . The expected reward is then given byk 1 2 K

c [ P c q P c q ??? qP cn 1 n1 2 n2 K n K

� 4Certainty. If we are sure that there is one c g 1, 2, . . . , K such that only the state s iscpresent, then we obviously have the value,

c [ cn nc

4.2. Examples

Let us now consider some simple examples in order to illustrate the methodproposed in the preceding section. For convenience, we fix in these examples the

w xsubset Y and the additive measure S taking Y s I and S A, B s B y A,;A F B g I. Accordingly, we omit the suffix S in the notation used above forthe total order between graded numbers, the corresponding minimum andmaximum, and the averages used to approximate the graded numbers by realintervals. Thus, the resulting notations are F for the order and min , maxl x l x l xfor the corresponding minimum and maximum. Also, any Zadeh’s graded

w Ž . w Ž . Ž .xxnumber c given by c a s a a , b a is approximated, when necessary, byw Ž . Ž .x lŽ .the compact interval AA c , BB c , or by the real number MM c , given by

1 1AA c s a a da BB c s b a daŽ . Ž . Ž . Ž .H H

0 0

1lMM c s lb q 1 y l a a daŽ . Ž . Ž .H

0

When applying the method proposed above, we also need to know thevalues of the levels a and a used in the ranking process. These values must ber s

Ž .determined according to their definition for each specific kind of gradednumbers. However, as follows, we shall do so for the triangular and for the

Ž .3-graded numbers see Definition 2.10 .

4.2.1. Rewards Gi en by Triangular Graded Numbers

Particularizing the preceding formulae to the triangular graded numberc s t , we immediately obtain thatŽ A, M , B .

A q M B q MAA s BB s

2 21lMM s lB q 1 y l A q MŽ .2


Ž . Ž .However, as a consequence, we find that: if AA t s AA t ,Ž A, M , B . Ž A9, M 9, B 9.Ž . Ž .BB t s BB t , and M s M9, then necessarily t s t .Ž A, M , B . Ž A9, M 9, B 9. Ž A, M , B . Ž A9, M 9, B 9.

In other words, the value M enables us to distinguish two triangular gradedw xnumbers which are approximated by the same interval AA, BB . Such a distinction

w xcan also be made, obviously, with the interval A, B . Therefore, we concludethat

a s 0 a s 1r s

With respect to the sum and product operations, it is easy to check thatt q t s tŽ A , M , B . Ž A9 , M 9 , B 9. Ž AqA9 , MqM 9 , BqB 9.

t if P G 0ŽPA , PM , PB .Pt sŽ A , M , B . ½ t if P F 0ŽPB , PM , PA.

Nevertheless, neither the min nor the max of two triangular graded num-bers must be another triangular graded number. To show this, it is suffices toconsider a simple example, such as the following one: t and t . Thus, inŽ0, 3, 4. Ž1, 2, 5.the first step considered above, it seems more convenient to use the min andl xthe max with these numbers.l x

Example. Let us consider a simple example with two alternatives, three states1 1 1Ž . Ž .of Nature s , s , s with respective probabilities , , , and the correspond-1 2 3 2 4 4

ing rewards given by

c s t c s t c s t11 Ž1 , 3, 4. 12 Ž1 , 2, 3. 13 Ž0 , 1, 2.

c s t c s t c s t21 Ž0 , 1, 3. 22 Ž0 , 2, 4. 23 Ž1 , 2, 3.

In these conditions, we expect to obtain the following rewards:

1 1 1v For the first alternative: c s t q t q t s t .1 Ž1, 3, 4. Ž1, 2, 3. Ž0, 1, 2. Ž3r4, 9r4, 13r4.2 4 4

1 1 1v For the second alternative: c s t q t q t s t .2 Ž0, 1, 3. Ž0, 2, 4. Ž1, 2, 3. Ž1r4, 3r2, 13r4.2 4 4

Now, we must rank the graded numbers c and c . We observe that1 2� 4c F c and therefore it must be c F c for all l g I and for all x g r, s .2 1 2 l x 1

Indeed, the expected rewards c and c are approximated, respectively, by the1 211 7 19 7 19 11w x w xintervals 3, and , . However, F 3 and F thus resulting that2 4 4 4 4 2

� 4c F c , ;l g I, ; x g r, s . Therefore, we choose the first alternative.2 l x 1

4.2.2. Rewards Gi en by 3-Graded Numbers

Now by particularizing the general formulae used above to the 3-gradednumber c s j , we easily obtain the following,Ž A, A9, A0 , B 0 , B 9, B .

A q A9 q A0 B q B9 q B0AA s BB s

3 31lMM s l B q B9 q B0 q 1 y l A q A9 q A0Ž . Ž . Ž .3


In the cases where the comparison between the 3-graded numbers j s1j X Y Y X and j s j X Y Y X moves to select a level a g I, weŽ A , A , A , B , B , B . 2 Ž A , A , A , B , B , B .1 1 1 1 1 1 2 2 2 2 2 2

have

1 w x w xif A , B / A , B1 1 2 23a sr 2½ w x w xif A , B s A , B1 1 2 23

w Y Y x w Y Y x1 if A , B / A , B1 1 2 2a s Y Y Y Ys 2½ w x w xif A , B s A , B1 1 2 23

With respect to the operations computed with the 3-graded numbersj s j , j s j X Y Y X , and j s j X Y Y X , it isŽ A, A9, A0 , B 0 , B 9, B . 1 Ž A , A , A , B , B , B . 2 Ž A , A , A , B , B , B .1 1 1 1 1 1 2 2 2 2 2 2

immediately possible to obtain that

j q j s j X X Y Y Y Y X X1 2 Ž A qA , A qA , A qA , B qB , B qB , B qB .1 2 1 2 1 2 1 2 1 2 1 2

j if P G 0ŽPA , PA9 , PA0 , PB 0 , PB 9 , PB .Pj s ½ j if P F 0ŽPB , PB 9 , PB 0 , PA0 , PA9 , PA.

� 4 X X Y Y Y Y X Xmin j , j s j1 2 Žmin� A , A 4 , min� A , A 4 , min� A , A 4 , min�B , B 4 , min�B , B 4 , min�B , B 4.1 2 1 2 1 2 1 2 1 2 1 2

� 4 X X Y Y Y Y X Xmax j , j s j1 2 Žmax� A , A 4 , max� A , A 4 , max� A , A 4 , max�B , B 4 , max�B , B 4 , max�B , B 4.1 2 1 2 1 2 1 2 1 2 1 2

Using these results, we can straightforwardly apply the two steps needed tochoose the best alternative in any decision problem with 3-graded numbers asrewards. To conclude, let us consider a simple example.

� 4Example. Let us assume that we have two alternatives v , v , the total1 2� 4uncertainty about the two possible states of Nature s , s and the correspond-1 2

ing rewards given by

c s j c s j11 Ž0 , 3, 6, 7, 10, 12. 12 Ž4 , 8, 9, 9, 10, 10.

c s j c s j21 Ž4 , 4, 6, 7, 8, 9. 22 Ž2 , 7, 9, 9, 11, 12.

The uncertainty about S gives rise to the introduction of subjective ele-ments in order to solve our problem. Thus, let us consider a degree of optimism

3l s and let us fix the parameter x s r.4

Under these assumptions, we follow Hurwicz’s criterion, thus obtaining thefollowing rewards associated with the respective alternatives,

3 1c s j q j s j1 Ž4 , 8, 9, 9, 10, 12. Ž0 , 3, 6, 7, 10, 10. Ž3 , 27r4, 33r4, 17r2, 10, 23r2.4 4

3 1c s j q j s j2 Ž4 , 7, 9, 9, 11, 12. Ž2 , 4, 6, 7, 8, 9. Ž7r2, 25r4, 33r4, 17r2, 41r4, 45r4.4 4

In order to rank these graded numbers, we approximate them by thew xcorresponding intervals, which are called AA, BB above. However, in this exam-

w xple, such interval is 6, 10 for both graded numbers. Therefore, we must select


1 w x wthe level a s to rank c and c . Thus, we consider the intervals A , B s 3,r 1 2 1 1323 7 45x w x w xand A , B s , . Calculating the corresponding average we finally2 22 2 4

150 75 1493r4 3r4obtain the values M s s and M s . Therefore the first alterna-1 216 8 16

tive is preferred to the second one.

3Remark. In the preceding example, the value l s is precisely the degree of4w xoptimism which produces the same interval AA, BB for the two graded numbers

c and c . For other values of the parameter l, the ranking process will be1 2easier.

For example, let us now take l s 0. In this case, we use Wald’s criterion,obtaining

c s j c s j1 Ž0 , 3, 6, 7, 10, 10. 2 Ž2 , 4, 6, 7, 8, 9.

Now these graded numbers are approximated, respectively, by the intervalsw x w x w x w xAA , BB s 3, 9 and AA , BB s 4, 8 . However, we consider first the left ex-1 1 2 2treme of these intervals because l s 0. Therefore, under these pessimisticassumptions, the second alternative is considered the best one.

This work is supported by the DGICYT under project PB95-1181.

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Solving a decision problem with graded rewards