Fuzzy Probability Spaces and Their Applications in Decision Matrices

Fuzzy Probability Spaces and Their Applications in Decision MatricesJana Talaˇs ov´a,Ondˇr ej Pavlaˇc kaDepartment of Mathematical Analysis and Applied MathematicsFaculty of Science,Palacky University Olomouce-mails:talasova@inf.upol.cz,onpav@centrum.czAbstractIn this paper,two types of fuzzy probability spaces will be introduced and their possible applications in methods of decision-making under riskwill be described.First,a fuzzy probability space that generalizes the clas-sical probability space( n,B n,p)to the situation of fuzzy random eventswill be studied.It can be applied e.g.when given continuous probabil-ity distributions of risk factors are to be approximated by discrete ones.Second,a fuzzy probability space that enables an adequate mathematicalmodelling of expertly set uncertain probabilities of states of the world willbe defined.The theoretical results will be illustrated with two examplesconcerning stock yields.1IntroductionIn decision-making under risk,decision matrices describe how consequences of alternatives depend on the fact which of possible and mutually disjoint states of the world occurs.The states of the world are supposed to be set exactly and their probabilities to be known.However in practice,the states of the world are often specified only vaguely and their probabilities are based on experts’estimations. In some cases,the states of the world and their probability distribution are obtained as a result of discretization of continuous probability distributions of risk factors.In this paper,it will be shown how the apparatus of fuzzy sets, especially two types of fuzzy probability spaces,can be used in such cases.2Basic NotionsA fuzzy set A on a universal set U,U=∅,is uniquely determined by its membership function A:U→[0,1].A set SuppA={x∈U|A(x)>0}is called a support of A.Sets Aα={x∈U|A(x)≥α},α∈(0,1],are called α-cuts of A.A set KerA={x∈U|A(x)=1}is a kernel of A.A fuzzy set AFuzzy probability spaces and their applications in decision matrices 141is called normal if Ker A =∅.The family of all fuzzy sets on U is denoted by F (U ).A normal fuzzy set A on the set of all real numbers ,whose α-cuts A α,α∈(0,1],are closed intervals,and whose support SuppA is bounded,is called a fuzzy number.The family of all fuzzy numbers is denoted by F N ( ).A fuzzy number A is said to be defined on [a,b ]if SuppA ⊆[a,b ].The family of all fuzzy numbers on [a,b ]is denoted by F N ([a,b ]).In this paper,trapezoidal fuzzy numbers will be used.The membership function of any trapezoidal fuzzy number A is piece-wise linear,uniquely determined by four points (a 1,0),(a 2,1),(a 3,1),(a 4,0);a 1,a 2,a 3,a 4are called significant values of A .A centre of gravity of a fuzzy number A defined on [a,b ]is a real number c A ∈[a,b ]given by the following formula c A = b a A (x ).x dx b aA (x )dx .(1)A fuzzy scale on [a,b ](see [4])is a finite set of fuzzy numbers A 1,A 2,...,A n that are defined on [a,b ]and form a fuzzy partition on [a,b ],i.e. n i =1A i (x )=1holds for any x ∈[a,b ].If a fuzzy scale represents a mathematical meaning of a natural linguistic scale,then it is called a linguistic fuzzy scale.A linguistic approximation of a fuzzy number A on [a,b ]by means of a set of linguistic terms {T 1,T 2,...,T s },where meanings of T k are fuzzy numbers T k on [a,b ]for k =1,2,...,s ,is the linguistic term T k 0,k 0∈{1,2,...s },such thatP (A,T k 0)=max k =1,2,...,n P (A,T k ),(2)where P (A,T k )=1− ba |A (x )−T k (x )|dx ba (A (x )+T k (x ))dx ,k =1,...,s.(3)Fuzzy numbers V i ,i =1,2,...,m ,defined on [0,1]are called normalized fuzzy weights (see [2])if for all α∈(0,1]and for all i ∈{1,2,...,m }the following holds:for any v i ∈V iαthere exist v j ∈V jα,j =1,2,...,m ,j =i ,such that v i +m j =1,j =iv j =1.(4)A fuzzy weighted average (see [2])of fuzzy numbers U i ,i =1,2,...,m ,on[a,b ]with normalized fuzzy weights V i ,i =1,2,...,m ,is a fuzzy number U on[a,b ]whose membership function is for any u ∈[a,b ]given by the formulaU (u )=max {min {V 1(v 1),...,V m (v m ),U 1(u 1),...,U m (u m )}|| m i =1v i ·u i =u, m i =1v i =1,v i ≥0,u i ∈[a,b ],i =1,...,m }.(5)The following denotation will be used for the fuzzy weighted average:U =(F )mi =1V i ·U i .(6)142J.TALA ˇSOV ´A,O.PAVLA ˇCKA 3Fuzzy Discretizationof Continuous Risk FactorsA decision matrix represents a suitable instrument of a risk analysis when risk factors affecting the consequences of alternatives are of discrete nature,moreover each of them takes a relatively small number of values.Then states of the world are given by all possible combinations of the risk factors values.If the considered risk factors are independent,then probabilities of states of the world are equal to the products of probabilities of the risk factors values.If they are dependent,then it is necessary to deal with their conjugate probability distribution (probabilities of states of the world are calculated as products of conditional probabilities of the risk factors values).In practice,the decision matrix method is applied also in cases when some of the risk factors are continuous.Then their continuous probability distributions have to be approximated by discrete ones.The approximation is more realistic when continuous domains (intervals)of the risk factors are replaced by fuzzy scales instead of real discrete scales.For that purpose,it is necessary to generalize the classical probability space ( n ,B n ,p ),where is the set of real numbers,B n is the σ-algebra of all Borel subsets of n ,and p is a probability measure defined on B n ,to the situation of fuzzy events.The corresponding fuzzy probability space will be defined as a triple ( n ,F B ( n ),P ),where F B ( n )is the family of all the fuzzy sets on n whose membership functions are Borel measurable,and the probability P (A )of any fuzzy event A ∈F B ( n )is given by the following formulaP (A )=nA (x )dp (x ).(7)It can be proved (see [1])that FB ( n )has the following properties:a) n ∈F B ( n ),b)if A ∈F B ( n ),then A ∈F B ( n ),c)if A i ∈F B ( n ),i =1,2,...,then ∞i =1A i ∈F B ( n );it means F B ( n )is a σ-algebra of fuzzy sets on n .Moreover,the mapping P :F B ( n )→[0,1]given by (7)satisfies the classical axioms of probability (see [1]).Therefore,it is meaningful to call the triple ( n ,F B ( n ),P )a fuzzy probability space.And evidently,if a set A belongs to B n ,then it belongs also to F B ( n )as a fuzzy set,and P (A )=p (A ).So,( n ,F B ( n ),P )is an extension of ( n ,B n ,p ).Now,it will be shown how the fuzzy probability space can be applied to a fuzzy discretization of continuous risk factors in decision-making under risk.First,let us suppose that consequences of alternatives are affected by only one continuous risk factor whose probability distribution is given by a density function f (x ).Let us define a fuzzy scale A 1,A 2,...,A n on the domain of the risk factor.The elements of the fuzzy scale are fuzzy random events,that is why their probabilities P (A i ),i =1,2,...,n ,are given by (7),i.e.P (A i )=SuppA iA i (x )f (x )dx.(8)Fuzzy probability spaces and their applications in decision matrices143It was proved(in[6])that ni=1P(A i)=1and P(A i)≥0,i=1,2,...,n;so,a discrete probability distribution on the elements of the fuzzy scale is defined by this procedure.If the consequences of alternatives are affected by several independent con-tinuous risk factors,then analogous procedure of fuzzy discretization is applied to each of them.All combinations of the risk factors fuzzy values determine states of the world,and probabilities of the states are given as products of probabilities of the particular risk factors fuzzy values.Now,let us suppose that the consequences of alternatives are affected by several(e.g.two)mutually dependent continuous risk factors whose conjugate probability distribution is given by a density function f(x,y).Then,fuzzy scales A1,A2,...,A n and B1,B2,...,B m will be defined on domains of both the risk factors.The Cartesian products A i×B j,i=1,2,...,n,j=1,2,...,m,where(A i×B j)(x,y)=A i(x)·B j(y)for any x,y∈ ,form a fuzzy partition on the Cartesian product of domains of both the risk factors.A conjugate discrete probability distribution of the fuzzy discretized risk factors is given by the following probability functionP(A i,B j)=Supp(A i×B j)(A i×B j)(x,y)f(x,y)dxdy,(9)for i=1,2,...,n,j=1,2,...,m.Possible states of the world are determined by such combinations of the risk factors fuzzy values for which P(A i,B j)>0.From the general point of view,the described fuzzy discretization means the replacement of a continuous random variable by a discrete fuzzy random variable with a specialfinite structure of(linguistically described)fuzzy values. Characteristics of the discrete fuzzy random variable are fuzzy numbers(or fuzzy vectors in the multidimensional case)calculated by means of the extension principle according to the classical formulas.E.g.the fuzzy expected value and the fuzzy standard deviation of a fuzzy random variable Z that takes values A i of a given fuzzy scale with probabilities P(A i),i=1,2,...,n,are defined by the following formulasF EZ=ni=1P(A i)·A i,FσZ=ni=1P(A i)·(A i−F EZ)2.(10)A similar probability distribution defined on a given linguistic fuzzy scale can be derived also directly from measured data.If measurements z1,z2,...,z m of a random variable Z are given,[c,d]is a domain of Z,fuzzy numbers A1,A2,...,A n form a fuzzy scale on[c,d],and m n,then probabilities of the fuzzy scale elements can be set by the formulaP(A i)=1·mj=1A i(x j),i=1,2,...,n.(11)144J.TALAˇSOV´A,O.PAVLAˇCKA This kind of probability distribution can be used not only in fuzzy decision matrices and in other fuzzy methods of decision-making under risk(fuzzy expert systems),but also as a general form of data presentation(like histogram etc.), that corresponds well to the human way of thinking.Example1:The fuzzy discretization procedure will be illustrated with an example con-cerning stock yields.Let us suppose that yields(in%)on stocks A and B are continuous random variables with normal probability distributions,for the stock A with parametersµA=4,σA=7.5,for the stock B with parameters µB=6.5,σB=12(see Fig.1).Figure1:Probability densities of yields on stocks A and BOn the closed interval of stock yields values coming in force,[−50,50],the linguistic fuzzy scale{NH,NVL,NL,NM,NS,AZ,PS,PM,PL,PVL,PH}is defined(see Fig2and Tab1).Probabilities of thefirst element(Negative Huge -NH)as well as of the last one(Positive Huge-PH)are equal to zero for both the stock yields,so the scale elements are not mentioned in Fig.2and Tab.1.Figure2:Linguistic fuzzy scale for the variable”Stock Yield”Discrete probability distributions on the given linguistic fuzzy scale are calcu-lated for the yields on stocks A and B by means of the formula(7)and are displayed in Tab.1.Fuzzy probability spaces and their applications in decision matrices 145ProbabilityLinguistic description Fuzzy Stock Yield (%)A BNegative Very Large (NVL)-37,5-35-25-22,500,01Negative Large (NL)-25-22,5-15-12,50,010,03Negative Medium (NM)-15-12,5-7,5-50,080,11Negative Small (NS)-7,5-5-2,500,160,11Approximately Zero (AZ)-2,5002,50,110,07Positive Small (PS)02,557,50,260,16Positive Medium (PM)57,512,5150,290,27Positive Large (PL)12,51522,5250,090,17Positive Very Large (PVL)22,5253537,500,07Table 1:Discrete probability distributions of fuzzy stock yields of A and B Fuzzy expectedvalues and fuzzy standard deviations of the yields on the stocks A and B are calculated according to the formulas (10);significant values of the fuzzy numbers are displayed in Tab.2,their graphic representations aregiven by Fig.3and Fig. 4.Let us notice that the original crisp expected values and standard deviations of the considered stock yields belong to kernels of the corresponding fuzzy expected values and fuzzy standard deviations.Fuzzy Stock Yield(%)F EA -0,352,155,808,30F EB 1,283,788,6511,5F σA 2,884,9911,3315,85F σB5,098,0216,6621,16Table 2:Fuzzy expected values and fuzzy standard deviations.Figure 3:Grafic representationFigure 4:Grafic representation of F EA and F EB of F σA and F σBThe linguistic approximation defined by (2)and (3)makes it possible to ex-press the main characteristics of the stock yields probability distributions also linguistically (see Tab.3).146J.TALA ˇSOV ´A,O.PAVLA ˇCKA Fuzzy numberLinguistic Approximation F EAPS F EBPS or PM [F EA −F σA,F EA +F σA ]NS or AZ or PS or PM [F EB −F σB,F EB +F σB ]NS or AZ or PS or PM or PLTable 3:Linguistic approximation of resultsComing out from the results of linguistic approximation,we can say,that the stock B is better than the stock A with respect to the criterion of yield.4Expertly Defined Fuzzy Probabilitiesof States of the WorldNot in all cases,when the decision matrix method is applied,probabilities of states of the world mean results of an exhaustive mathematical risk analysis.Especially if states of the world are specified vaguely or if they are affected by a large number of hardly describable risk factors,their probabilities are set only on the basis of experts’knowledge and experience.To enable a correct mathematical modelling of uncertain probabilities of a finite set of random experiment results (states of the world in decision matrices),it is necessary to extend the classical probability space (Ω,P (Ω),P ),where Ω={ω1,ω2,...,ωr },P (ωi )=p i ,p i >0,i =1,2,...,r , r i =1p i =1,P (Ω)is the family of all subsets of Ω,and P (A )= {i :ωi ∈A }p i for any A ∈P (Ω),to the situation of fuzzy probabilities of elementary events.Uncertain expertly set probabilities of elementary events ω1,ω2,...,ωr will be modelled by normalized fuzzy weights P 1,P 2,...,P r .The fuzzy probability of any random event A ∈P (Ω)will be defined by means of the fuzzy weighted average operation in the following wayP (A )=(F )ri =1P i ·χA (ωi ),(12)where χA (.)is the characteristic function of the set A .Analogously,the fuzzy probability of any fuzzy random event A ∈F (Ω)will be defined by the formulaP (A )=(F )ri =1P i ·A (ωi ),(13)where A (.)is the membership function of the fuzzy set A .A fuzzy probability space with a finite set of elementary events is defined as a triple (Ω,F (Ω),P ),where Ω={ω1,ω2,...,ωr }is a set of elementary events whose fuzzy probabilities are given by normalized fuzzy weights P 1,P 2,...,P r ,P i =0for i =1,2,...,r ,F (Ω)is the family of all fuzzy sets on Ω(fuzzy random events),and a mapping P :F (Ω)→F N ([0,1])assigns to each fuzzy random event A ∈F (Ω)its fuzzy probability P (A )according to (13).Fuzzy probability spaces and their applications in decision matrices 147It can be proved (see [6])that the mapping P has properties representing a generalization of axioms of the classical probability:a)P (Ω)=1,b)∀A ∈F (Ω):P (A )≥0,c)for any A 1,...,A s ∈F (Ω),such that A i ∩A j =∅for i =j :P ( s j =1A j )=(F )(P (A 1)·1+P (A 2)·1+...+P (A s )·1+P (A 1∪...∪A s )·0).Similarly,the family of all fuzzy sets on Ωforms a σ-algebra of fuzzy sets.So,it is really meaningful to call the triple (Ω,F (Ω),P )a fuzzy probability space.In a fuzzy probability space (Ω,F (Ω),P )with a finite set Ω,any mapping U :Ω→F N ( ),where F N ( )denotes the set of all fuzzy numbers,defines a discrete fuzzy random variable.E.g.fuzzy evaluations of an alternative under states of the world whose fuzzy probabilities are known represent a discrete fuzzy random variable.The probability distribution of a discrete fuzzy random variable U is given by a mapping P (U i )=P i ,i =1,2,...,r ,where U i =U (ωi )and P i =P (ωi )for i =1,2,...,r .An expected fuzzy value F EU of a discrete fuzzy random variable U is defined as a fuzzy weighted average of fuzzy values U 1,U 2,...,U r with fuzzy probabilities P 1,P 2,...,P r ,i.e.F EU =(F )ri =1P i ·U i .(14)5Fuzzy Decision MatricesLet us consider a problem of decision-making under risk that is described by the following decision matrix,where x 1,x 2,...,x n are alternatives,S 1,S 2,...,S r states of the world,P 1,P 2,...,P r their fuzzy probabilities,and U i,k ,i =1,2,...,n ,k =1,2,...,r mean fuzzy degrees in which the alternative x i satisfies the given decision objective if the state S k occurs:S 1S 2...S r P 1P 2...P r F EU x 1U 1,1U 1,2...U 1,r F EU 1x 2U 2,1U 2,2...U 2,r F EU 2..................x nU n,1U n,2...U n,r F EU n Table 4:Fuzzy decision matrixFuzzy numbers F EU i express expected fuzzy evaluations of alternatives x i for i =1,2,...,n ;it means they are calculated according to the formulaF EU i =(F )rk =1P k ·U i,k .(15)The best of the alternatives will be chosen by the rule of the maximum expected fuzzy evaluation.For that purpose some of the preference relations on F N ( )(see [4]),e.g.the ordering of fuzzy numbers according to their centres of gravity148J.TALA ˇSOV ´A,O.PAVLA ˇCKA can be applied;also a linguistic approximation of the expected fuzzy evaluations of alternatives by means of an ordered set of evaluating linguistic terms can be used.A similar approach can be applied also to multi-criteria decision-making under risk.In [5],utilization of expertly defined fuzzy probabilities in three-dimensional decision matrices is presented,where multi-criteria evaluating pro-cedures are based either on fuzzy weighted averages of partial fuzzy evaluations or on fuzzy expert systems (see also [3],[4],[2]).Example 2:Let us consider a problem of comparing two stocks,C and D ,with respect to their future yields.The starting prices of C and D are 2900and 3300CZK,respectively.An expert estimates probabilities of future states of the economy (economic drop ,economic stagnation and economic growth )by fuzzy numbers P 1,P 2,P 3whose significant values have to satisfy the natural conditions 3i =1p 1i ≤1, 3i =1p 2i ≤1, 3i =1p 3i ≥1and 3i =1p 4i ≥1,but need not necessarily represent normalized fuzzy weights.The fuzzy proba-bilities P 1,P 2,P 3are obtained from P 1,P 2,P 3by the following transformation p 1i =max {p 1i ,1− 3j =1,j =i p 4j },p 2i =max {p 2i ,1− 3j =1,j =i p 3j },p 3i =min {p 3i ,1− 3j =1,j =i p 2j },p 4i =min {p 4i ,1− 3j =1,j =i p 1j },where i =1,2,3,which eliminates a possible inconsistency of experts estima-tions.The future stock prices of C and D under the given states of the economy are also estimated by fuzzy numbers.Corresponding fuzzy stock yields (in %)are calculated by means of the extension principle according to the formula yield =new price −old price old price·100.Economic Drop Econ.Stagnation Economic Growth P 0,10,150,250,30,40,50,60,750,20,30,350,55P0,10,150,20,30,40,50,550,70,20,30,350,5C-price220024002600280027502900300031503100325034003600C-yield-24,1-17,2-10,3-3,5-5,203,58,66,912,117,224,1D-price240025502750290030003150350037003750390041004300D-yield -27,3-22,7-16,7-12,1-9,1-4,66,112,113,618,224,230,3Table 5:Fuzzy decision matrix for stocks C and DFuzzy probability spaces and their applications in decision matrices149Exp.Fuzzy Stock Yield Centre of GravityF EC-8,450,176,2115,173,29F ED-10-1,379,0118,794,15Table6:Expected fuzzy stock yields,their centres of gravityExpected fuzzy stock yields presented in Fig.6and Tab.6were calculated by the formula(15).Fuzzy numbers F EC,F ED are incomparable in the sense of the usual ordering of fuzzy numbers that is based on the ordering of their α-cuts.But as the centre of gravity of F ED is greater than that one of F EC, the stock D seems to be better than C with respect to the criterion of yield.6ConclusionModels of decision-making under risk are more realistic when the mathematical apparatus of the fuzzy sets theory is applied.The fuzzy discretization of con-tinuous risk factors is better than the classical discretization using scales of real numbers.Similarly by setting probabilities of states of the world,the exploita-tion of experts’knowledge is higher when fuzzy probabilities are used instead of the classical real ones.References[1]Negoita,C.V.,Ralescu,D.A.(1975),Applications of fuzzy sets to systemsanalysis.Stuttgart,Birkhuser Verlag-Edituria Technica1975.[2]Pavlaˇc ka,O.(2004),Normalized fuzzy weights.(In Czech.)Diploma thesis.Faculty of Science,Palacky University,Olomouc2004.[3]Talaˇs ov´a,J.(2000),NEFRIT–Multicriteria decision making basedon fuzzy approach.Central European Journal of Operations Research, Vol.8(4)-2000,297-319.[4]Talaˇs ov´a,J.(2003),Fuzzy methods of multicriteria evaluation and decision-making.(In Czech.)Publishing House of Palacky University,Olomouc2003.[5]Talaˇs ov´a,J.(2005),Fuzzy sets in decision-making under risk.(In Czech.)Proceedings of the4th Conference Aplimat,Bratislava,February1-4,2005, 532-544.[6]Talaˇs ov´a,J.,Pavlaˇc ka,O.(2005),Fuzzy methods of decision-making underrisk.(In Czech.)Unpublished technical report.Faculty of Science,Palacky University,Olomouc2005.。