Kα'gaωa Universify Ewnomic Review Vol.73， No 2， September2000， 111-140
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Abstrαct: In thisραρeη the bαsiじ，conceptsofρossibiliわJthe -oηαre briefiy introduced. Ajter that， theU.ρρer and lower ρossibility distributions are identified to characlerize the knowl -edge斤.omone ex.ρer.t Multi-source knowledge斤ommultiple ex;ρerts is represented by a set of ex沙onenti(uρossibiliか， distribu -tions.. Based on the consistenιy index defined byρ
ossibiliか， measure of eachρair ofρossibili.かめのstributions; theル
sion model isρroposed to integrate the multitleρossibili砂 diStribu -tions into a new one1'<ψresenting a refined knowledge. Lastly theρortjolio selectionρroblem with multitle e;xρerts is consid -ered 1. Introduction The remarkable advance of computer techniques has brought about a present-day information age characterized by the acceleration， intellectual -ization and globa1ization of information， which has stimulated a more emergent requirement for dealing with the huge and sophisticated informa -tion in the real world. Knowledge representation and fusion based on possibility theory is one of newly-emerging information techniques to intelligently deal with human knowledge for meeting such needs [1， 3， 14， 18J ..112ー Kagawa University E.正onomicReview 370
Generally speaking the vagueness and ambiguity of human understand -ing， the ignorance of cognition and the diversity of evaluation are always contained in human knowledge“ A possibility distribution is a kind of representation of knowledge and information where the center reflects the most possible case and the spread reflects the others with relatively low possibilities The area of the possibility distribution can be regarded as a sort of measure of fuzziness In some case， it is difficult to directly give some possibility distribution to represent expert knowledge However， we can always obtain the possibility grades of discrete data from an expert
児 島ctinghis judgement on some specified event For example， in portfolio
selection problems， experts can choose some typical patterns from the past security data and give them associated possibility grades to児 島cttheir judgment about the situation of stock markets in the future. The higher the possibility grades of security data， the more similar to the future.
In this paper， the dual possibility distributions， called upper and lower possibility distributions， are defined with considering the concept of rough sets to characterize the intrinsic uncertainty of human judgment [9， 10， 13J These two possibility distributions are identified from the given data to reflect two extreme opinions on a specified event， for example， the predic -tion of stock markets. The upper possibility disttibution can be regarded as an optimistic viewpoint and the lower distribution as a pessimistic one in the sense that the upper possibility distribution always gives a higher possibility grade than the lower one
Information fusion is a mathematical method to obtain a refined infor -mation from multiple information sources， which conflict with each other in nature， such as multiple sensors and multi-expert pooL Generally speak -ing， the information fusion models can be built based on probability and possibility theories， respectively， to represent the intrinsic uncertainty
371 Decision Analysis Based on Possibility Theory 113
contained in multi-source information The probability networks， such as Bayes networks and Markov networks are well-known probability methods for information fusion where the information is presented as a conditional probability distribution and the fusion procedure is based on Bayes formula [l1
J
Dempster-Shafer theory of evidence (DS) is an important tool of information fusion to deal with the non-additional probability phenomena where the fusion procedure is based on the Dempster's rule of combination [12J.. Dubois， Prade and Yager proposed some information fusion models based on possibility theory [1， 2， 18， 19J The approaches related to infor -mation fusion for decision analysis have been researched in the papers [4， 5， 6， 7J In this paper， a set of exponential possibility distributions is used to characterize the knowledge of multiple experts.. The consistency index of two possibi1ity distributions is defined based on possibility measure to reflect their similarity degree The possibii1ty distributions are preproces -sed so that the possibility distribution with the lower consistency index with the other possibility distributions is regarded as ou日erto be eliminated. A fusion model is proposed to fuse the multiple possibility distributions into a new possibility distribution representing a more reliable knowledge.. In order to do that， the paper is organized as follows In Section 2， the most common axiomatic characterizations of possibility theory and some basic properties of possibility and necessity measures are introduced. In Section 3， the concepts of upper and lower possibi1ity distributions are introduced The identification method is proposed to obtain the dual possibility distributions In Section 4， the fusion of multisource knowledge is considered where each knowledge source is characterized by an exponential possibility distribution. Lastly the portfo1io selection problem with a group of experts is considered in the section 5.. In section 6， a-114 Kagawa University Economic Review 372 numerical example is given to show the proposed methods.. Finally， some concluding remarks are included
2
.
Possibility theoryPossibi1ity theory is one of several formal mathematical systems that are suitable for characterizing and analyzing the uncertainty of various types Possibility theory has the very close relation with probabi1ity measure theory and fuzzy measure theory. In order to understand possibil -ity theory easily， let us begin with probability measure theory Given a universal setQ and
r
is a σfield on Q， probability measure f is the mapping as follows f:r
→[0， 1]， that satisfy the following requirements : (f 1)P(φ) = 0; (f 2)P(Q) = 1 ;(f3) For any A E
r
and A; Er
， ifAi
n
A;=φ(zヰ j，z， j = 1， 2，…，)→P(Ui=l Ai) =L
:
i=l P(Ai)(Ad-ditivity) A fuzzy measure g is a mapping as follows g:r→[0， 1]， that satisfies the following requirements : (g 1)g(φ)=
0; (g 2)g(Q) = 1 ; (g 3) for all A and B Er
， ifA三B，theng(A)ζ g(B) (monotonicity)It can be seen that fuzzy measure is a generalization of probability measure for dealing with the non-additivity cases where the additivity is loosen to be monotomcity..
373 Decision Analysis Based on Possibility Theory measure and necessity measure defined below ..
A possibility measure， Pos， is a function Pos: r→[0， 1]， that satisfies the following requirements : (Pos1)Pos(φ) = 0; (Pos2)Pos(Q) = 1 ;
115-(Pos3)for any family{AIAi E r， i E I}， where 1 is an arbitrary index set， Pos (
出
Az)=tppos(Az) A necessity measure， Nec， is a function Nec:r
→[0， 1]， that satisfies the following requirements : (Nec 1)Nec(φ) = 0; (Nec 2)Nec(Q) = 1 ;(Nec 3)for any family{AilAiε r， i εI}， where 1 is an arbitrary index set， Nec(内Ai)= infNec(Ai)
iel iel
It can be seen that the possibility measure is the lower semicontinuous fuzzy measure (for any increasing sequenceAl c;;A2 c;;…of sets inr， ifUi=lAi E r， then limiド司叩∞g(Ai)= g(Ui主二
s
詑emic
∞
ontinuousfuzzy measure (for any decreasing sequenceAl :2A2 ;ヨ2"… 削 川"0ぱfs
記etぬsin r， ifnAiEr， then limぃ"，g(A)= g(ni=lAi))..
The following dual relation between possibility measure and necessity measure holds
Nec(A) = 1-Pos(A C)， (1) that means that based on the formula (1)， giving either of the definitions of possibility measure and necessity measure can lead to the other.. It is obvious that based on (1)， (Pos1)and (Pos 2) can lead to (Nec 2) and (Nec 1)， respectively. Let us now check the case ofA =内Ai In this case， 1
116'- Kagawa University Economz( Revieω 374 Pos(AC) = 1-pos((nAi)C) = 1-Pos(UAf) = l-sup Pos(Af) = inf(l -Pos(Af))holds from (Pos3)，inf(1-Pos(Af))=inf iv叫 ん )and 1 -Pos(AC) = Nι(A) hold from (1) so thatNec(
n
Ai) = infNec(A，) that is (Nec 3) Likewise， it is true that based on (1)， (Nec 1)， (Nec 2) and (Nec 3) can lead to (Pos 1)， (Pos 2) and (Pos 3) Definition1. Given a function ァ: X→[0，1] 1 i 一 一 、 、 s ， J ゲ ん 〆 ， ， ‘ 、 7 p x H u e s x z i ・ 1 then the function r is called the possibility distribution ofX
It can be seen that the possibility distribution characterizes the unique possibility and necessity measure via the following formulas Pos(A) = supr(x)， (2) X E A El'{.X) Nec(A) = 1- sUj)..• r(x)， (3) X E Aじεl()(l r(x)= Pos({x}) xEA. (4) Give a possibility distributionIIA(X)and a fuzzy event (fuzzy set)B with the membership function μs(x ).. The definitions of possibility and necessity measures ofB based on IIAx) are as followsIIA(B)= sup{IIA(x)^μs(x)}， x
NA(B) = inf{(1-IIA(x))Vμs(x)}， x
(5) (6) Similarly， the iollowing dual relation between IIA(B)and NA(B) holds
NA(B) = 1一IIA(BC ) (7) (7) can be easily understand from the following transformation l一IIABC )= l-sup{IIA(x)^(l一μs(ェ))} x = inf{l-IIA(x)^(l一μs(x))} = inf{(l-IIA(x))Vμs(x)} = 民4(B)， (8)
-117 -where 1一α/¥b= (1-a) V (1-b) is used. LetX be a possibilistic variable governed by a possibility distribution Decision Analysis Based on Possibility Theory 375 Given an inequality relation X~z，
the possibility and necessity measures of X ~ z denoted as Pos(X 三二z) and
7rA..
Ne
，
(X三 z)，respectively， are obtained from (5)and (6)as follows :Pos(Xζ z)= sup{
ぬ
い
)Ix~ z}， Nes(X ~ z)= l-sup{πA(x)lx>z}..In the cases of Pos(X手 z)and Nes(X ~ z)， B is the crisp set (ーα，z)
The explanations are shown in the following figure.. (9) ( 10) Nes(Xζz)= 0 X 早向巾民 H U 巾 門 田 町 田 明 ) ︿ 白 戸 己 巾 (1) X Z X 三 冊 B U R S H 匂 ︿ 同 - 5 (2) N白(Xζ z) Pos(X:::;; z)= 1 X The possibility and necessity measures ofXζ z Z Fig.1
e l
i l
i l
i
-- !
118ー Kagawa UniversiかEconomicRevieω 376
3
.
Identification of Dual Possibility Distributionsfrom Given Data
3. 1 The concept of upper and lower possibility distributions
The knowledge from one expert can be represented by a data set{(Xi， h，
)
l
z
= 1，…， m} where x， = [九…，Xin]tis an n-dimensional vector to characterize some specified event， hi is an associated possibility gradegiven by an expert to refiect his judgement on how much the possibility grade of theithsample is for this event， and m is the number of samples The data set(Xi， hi) (i= 1，…， m) can be approximated by a dual data sets(Xi， hli) and (Xi， hUi) (i= 1， …， m) with the conditionhli三二 hi:S::.hUi'
Assume that the valueshliand hUiare from a class of the functionsG(.x， ())
with the parameter vector θLet G(x， ()l)and G(x， ()U)correspond tohli
and hUi(i= 1，…， m)， respectively and simply denote asiCl(X)and iCu(X)
Given the data set(Xi， hi) (i= 1，…， m)， the objective of estimation is to obtain two optimal parameter vectors ();; and ()t from the parameter spece to approximate(Xi， hi) from upper and lower directions according to some given measure， Moreover， the dual optimal parameter vectors (();;， ()t) make the relationG(x， ()t)ζ G(x， ();;) hold for any arbitrary n-dimen -sional vectorX Suppose that the functionG(x， ()) is an exponential function exp{一(X -α)tDA1(x一α)}，simply denoted as (α， DA)eド Thenthe following for -mulas hold πt(Xi)
=
exp{一(X，一α)tDi1(Xi一α)}=
α，( Dふ(i=
1，…，m)， (1I) π'U(Xi)=
exp{一(Xi一α)tD;;l(Xi一α)}=
α， ( Du)e(i=
1，…， m)， (12) ゎ(Xi)ζ hi :s::.iCu(Xi)and iC/X) 三二 iCu(X)， (13) where α = [al，ぬい。"'an]tis a center vector， Du and Dl are positive definite matrices， denoted asDu>
0 and D 1> 0， respectively， It can be377 Decision Analysis Based on Possibility Theory -11少ー
seen that in the above exponential function， the vector αand matrices Du and D 1 are parameters to be solved Di任erentparameters α， Du and Dl lead to different values7l"1(Xi) and 7fu(Xi) which approximate the given
possibility degreehi of Xi to the di任erentextent
Definition2. Given the formulas(11)， (12)and (13)， the fitness of approxima -tion based on the parameters α， Du and Dl， denoted ass， is defined as follows:
(14) It is known from Definition 2 that the higher the parameterβis， thec10ser tohi the values7f1(X;) and 7fU(Xi) are from lower and upper directions，
respectively
Definition3. Denote the optimal solutions ofα， Du and Dl as α*， D*u and D*l， respectively， which maximize βwith the constraint(13) The following functions π*1(X) = exp{一(X一α*YD*l(x-α*)}， πh(X) = exp{ ー (X- α*)tD*~(x α*)}， ( 15)
ω
are called lower and upper exponential possibi1ity distributions of the vector x， respectively For simplicity afterwards we write7fu(x) and 7f1(X)instead of7f*u(x) and 7f判(x) The concept of upper and lower possibility
distributions is il1ustrated in Fig 2. It can seen from Fig 2 that the giyen possibi1ity degrees are completely inc1uded by upper and lower possibi1ity distributions. The upper possibi1ity distribution can be regarded as an optimistic viewpoint and the lower distribution as a pessimistic one in the sense that the upper possibility distribution always gives a higher possibility grade than the lower one. The difference between the dual possibility distributions reftects the inconsistency of expert knowledge..
-12{}-- Kagaωa University Economic Revieω U Y Fig. 2 The concept of upper and lower possibility distributions (The upper curve is the upper possibility distribution and the lower curve is the lower possibility distribution)
3. 2 Identification of upper and lower possibility distributions
378
The model to identify the upper and lower possibility distributions can be formulated to maximize the fitness measure as follows.
。
力
sれt. 7r1(X，)ζ hi，
π'U(Xi)注 hi，
π'U(X)注 π/x)
With considering that maximizing
"
1
.
/
丹
空
完
斗
isequivalent to maximizing379 Decision Analysis Based on Possibility Theory iBist
呂
(Xi α)tDI1(xi一α)呂
(Xiー α)tD;/(Xi一α) s. t (Xi α)tDI1(xi α)ミ -lnhi (i= 1，… ， m)， (Xi一α)tD;l(X，一α)三 一lnhi (i= 1，… ， m)， Du-Dlミ0， DI>O 121-(18) In the following， let us consider how to obtain the center vector αand the positive matricesDl and Du. It is straightforward that upper and lower exponential distributions should have the same center vector.. Other -wise， the relation 7l"uCx)ミ 7r/X)can not hold when the vectorX is equiva -lent to the center of the lower possibility distribution Because the vector X with the highest possibility grade should be closest to the center vector α among allXμ =
1，… ， m)， the center vector αcan be approximately estimated as α= Xi*， (19)where Xi*denotes the vector whose grade ishi*= kmax =l. .mhk The associat
-ed possibility grade ofXi*is revised to be 1 because it becomes the center vector Taking the transformationy = x-α， the problem (18)is changed into the fol1owing problem m m q 苧11叩!1?:一!yd3イyμ
f
yD
l1仇3 Uu仙，Ult=l t=l s..t Y巴;/Dl1弘 3 D;lYi~ζζ 一 f如nhμ=1
，い紗…"川.， m)， ， Du-Dl 二三 0， Dl>
0 (20)The problem (20)is a nonlinear optimization problem due to the last two constraints To cope with this difficulty， we use principle component analysis(PCA) to rotate the given data(Yi， hi) to obtain a positive definite
380 Kagawa UniVersity EcvnomiじReview -122ー The data can be transformed by 1inear transformationT matrix easi1y Co1umns ofT are eigenvectors of the matrix~ = σz[;]， where σ;; is defined as 竹H H 一 9 ' u ( σ吋={呂(Xki-ai)(Xk;ーム)ん}/Elhh Using the linear transformation， the dataYiis transformed into{Zi= Then the formu1as (1)1and (12) can be rewritten as follows
幻(Zi)
=
exp{ -zfTtD;;l Tzi} (i=
1，…， m)，7rU(Zi)= exp{ -zfTtD;;l Tz，}(i
= 1，…， m)，
Here TtD;;lT and TtDi1T are assumed to be diagona1 matrices as fo1
-(22) (お) Tty，}" ( 24) (25) 10ws: I Cul 0 ¥ Cu = TtD;;lT =
I
，
1
¥o Cun/ I Cl1 0 ¥ C1 = TtDi1T =I
¥o Cln/ The mode1 (18) can be rewritten as the following LP prob1em :。
。
η m min ， zfC'2 品 -2， ZfC' uZi Cl.Cu i=l i=l s" t zfCUzi三三一lnhi， zfC品 二 三 一lnhμ=
1，…， m)， C(j二::::Cu，，i Cuj注 ε(j=
1，…， n)，Where the conditionCl;注 C町二三 ε>0makes the matrixDu-Dl semi
Denote the optima1 positive definite and matricesDu and Dl positive" Thus， we have solutions of (26) as
C
:
:
andc
r
。
司
D: = TC::-1Tt， Dt = TCt-1Tt381 Decision Analysis Bas巴don Possibility Theory
123-For simplicity afterwards we writeDu and Dl instead ofD~ and Dt Theorem 1. There always exists a functionL(x，θ) which can approxi -mate (Xi， hi) (i= 1，… ， m) from upper and lower directions.
Proof. SettingL(，x. 8) as the exponential function exp{一(x一α}tDA1(x -α)}， let us take Cu = qI and Cl =ρ1 in(26) Thus the constraint condi -tions of(26)can be written as ρIzfzzミ -lnhi，i = 1，… ， m， qzfz， ~ -lnhi， i = 1， ....， m， q 主主ε， ρ:?q (28) If we take ρ m a x i=l. *(-lnh
ムたよ
q= min *(-lnhdzfzi)and ε ，.m-l，iキ i=l，，.m-l，i本z 三 二 q， inequalities(28)can hold Therefore， there is an admissible set in the constraint conditions of the LP problem (26). It should be noted that the vectorZi*=
0 is omitted， becauseZi*tZi=
ln1=
0 in(:沼). Thus， we con -sideri=
1，… ， m-1 withoutZi*=
0 in determining the values for ρand q..口
Theorem 2. Assume that the fuzzy vector X is governed by an n-dimen -sional possibility distribution (α， DA)e， denoted asX ~(α， DA)e， the possi -bility distribution ofY with Y = rtX， denoted asiTB(Y)， is as follows: πs(y)= eゅ{一(y一戸α)2(rtDAr)-l，} (2。
where rtαis the center value and rtDAr is the spread value of Y Y~(rtα，rtDAr)eis called the one-dimensional realization ofX ~(α， DA)e
Proof. The possibility distribution of fuzzy number Y， denoted asゐ(y)can
be obtained by the extension principle as
πs(y)= max. exp{一(x-α}tDA1(x-α)} 側 {xly=r"x}
The optimization problem (30)can be reduced to the minimization problem of Lagrangian function as follows :
-124ー Kagawa University E.じonomicReview 382
L(x， k)= (x α)tD;;"l(X一α)+k(y-rtx) (3]) The necessary and sufficient conditions for optimality， that is， oL/ox
=
0，。
L/ok= 0， yield the following equations 2D;;，.1(X一α)-kr= 0， y-rtx =.
o
From (32)， x* is obtained as x* α+(k/2)DAr， Substituting(3)4into(33)leads to k*= 2(y-rta)/(rtDAr) Substituting(35)into(34)leads to x* α+(y一戸α)DAr/(rtDAr). Substituting(36)into(30)， we have πs(y)= exp{一(y-rtα)2/(rtDAr)} This proves Theorem 2.4
.
Models for Fusing the Knowledge from Multiple Experts。
2) (33) ( 34)。
5) (36) (37) 口 Given a data set{< (Xl， h，)l (X2， M)，…， (Xm， h~)> ， … ， <(Xl， M)， (X2， M)， .." '" (Xm， h~) >}， where Xk = [Xkl，…， Xkn] t is an n-dimensional vector， h~ is an given possibility grade by the ithexpert to reflect his judgement on how much the possibility grade of the kthsample is for some specified event， m is the number of samples and s is the number of experts Using the above mentioned identification method， s dual possibility distribu -tions can be obtained to児 島ctthe inherent diversity in human thought The set formed by these dual possibility distributions， denoted asU = {X ~<(αj， Dui)e， (αi， Dli)e>li= 1，… ， s}， is called an information block where X~<( αi ， DUi)e， (αi， Dlふ >is obtained from the data set {(Xl， h，)i (X2， M)，…， (Xm， h:n)} In the following we focus on how to fuse possibility383 Decision Analysis Based on Possibility Theory distributions in
U
to obtain a refined knowledge4. 1 Consistency indexes of multiple exponential possibility distributions
125
Definition 4. Given possibility distributionsx，~( αi ， Di)eand Xj~( αH
Dj)e， their consistency index， denoted asr(Xi， X
ふ
isdefined as follows， where r(X" Xj)= max(九 (.x).7rx，(x)) あ，(x)= exp{一(x-αYDi1(x一αi)}= (αi， Dム
πX，
(Xi)= exp{一(.x αj)tDj-l(X-αj)}= (αj，
D;)e (38)Theorem 3. Give po部ibilitydistributionsXA~( α， DA)eand XB~(b， DB)e， their consistency indexr(XA， XB) is as follows :
r(XA， XB)
=
exp{(DA1α十D;;lb)t(DA1+ Di1)-1(DAlα+Di1b)一αtDA1α-btDi1b} Proof.Itfollows from (38)that
r(XA， XB) = max{exp{一(X α)iDA1(X一α)}exp{一(.x-b)tDi1 (X-b)} (39) = max{exp{一(X-α)tDA1(X一α)一(X-b)tDi1(x-b)}，
旬
。
which leads to the following optimization problem : min f(x)= (x一α)tDA1(X-α)+(x-b)iDi1(.x-b) (41) Di任erentiating(41)with respect to，x.the optimum solutionx* isx*= (DA1+ Di1)-1(DAlα+Di1b) (42) Substituting.x*into(40)， we can obtain the consistency index as follows
r(XA， XB) = exp{(DA1α+ Di1 b)t(DA1
+
Bi1)ー1(DA1α十Dii1b)126 Kagawa UniversiかEconomicRevieω 384 It is clear from the definition of the possibility measure that 0 三二llA(B) ζ L In the information blockU
=
{X~ く (αi ， Dui)e， (αi， Du)e>
I
z
=
1， …， s}， we can calculate consistency indices of Xi andX
;
based on upper and lower possibility distributions， respectively which are called the upper and lower consistency indices ofXi and X; and denoted asrl(Xi， X;) andrU(Xi，
X
.
ふ
respectively.. The consistency indices ofU based on upper andlower possibility distributions， denoted asrU and rl， respectively are as follows rl = min rl(Xi， X;よ i，;E(l， ，8) rU = min rU(Xi， X;) i，;E(l， ，8) 制) 倒 Definition 5. Letゐandふbethe predetermined thresholds forrU and rl，
respectively. An information blockU is optimistically reliable if and only ifrU is not less than ou and its cardinality is not less than 2. An imforma -tion blockU is pessimistically reliable if and only ifrl is not less thanふand its cardinality is not less than 2.. An information block U is reliable if and only if it is both optimistically and pessimistically reliable. If an information block is not reliable， it should be preprocessed before fusion. The basic idea for preprocessing is to delete the outliers from the given information block. In other words， a possibility distribution， which has the lower consistency index with the others， can be regarded as an outlier to be deleted The following algorithm is used to obtain the reliable block from the given information blockU with the maximum cardinality， which is called as the efficient block ofU， denoted asUム
Step 1: The s upper and s lower consistency indices are represented by two s x s matricesQu
=
[qf;J and Ql=
[qf;] where qf;=
rU(X， X) and qfj=385 Decision Analysis Based on Possibility Theory 127 が
(X
，X)
(i= 1， . 叶 s;j = 1，…， .$) Step 2: The e1ementsqg(qfj)is transformed into 0 or 1 with the conditions that ifqgと δu(q;fミふ)thenqg = l(qfj= 1) e1seqg= O(qfj= 0).. Obtain a binary matrixQ* so thatQ* = [q口]sxs= [qg. qfj]. Step 3: Denote the index of the row ofQ* with the biggest numbers of the e1ement 1 asi* If the numbers of 1 ini*is 1arger than one， the e伍cient b10ck is obtained as the setUe = {Xj~< (αj， D，"j)e， (αj， D(j)e>Iqi判 =1， j = 1，吋 s}“ Otherwisether e is no e伍cientb1ock， then the information b10ckU is called confl.ict information b1ock. Theorem 4.U is a confl.ict information b10ck if and on1y if'¥ji， jε{1， ".， s}， iキj，ri~< (}Uor r/j< (}l山 Otherwise，there must be an efficient informa -tion b1ock. IfU is a confl.ict information b10ck then any subset ofU is a1so a confl.ict information b10ck 4. 2 Fusion model based on consistency index After obtaining the efficient b1ocks， 1et us consider how to fuse the possibility distributions in the obtained efficient b10cks Without 10ss of generality， suppose that there is on1y one e伍cientinformation b10ckUe ={X~<( αi， Dui)e， (αi， D
ル>
Ii= 1，…， ρ} obtained from U whereρis the cardina1ity ofUふ Thereliabi1ity degree of Xi~<( ι ， Dui)e， (αi， Dli)e> inUe， denoted as R(Xi) can be calcu1ated as follows R(X)= L
:
(rU(X， Xj)+ rl(X， Xj))(i = 1，…， ρ) 働 jE{I....pj'iキy According to the va1ues of R(Xふ
thedata set {< (Xl， hl)， (X2， hD，…， (Xm， h~)> ， …，<
(Xl， hf)， (X2， M)，…， (.Xm，紘)>} can be reordered as{
<
(Xl， hl*)， (X2， M*)，ー，(Xm， h~*) >，…， {< (Xl， M*)， (X2， M*)， ."， (Xm， h品*) >，… ， <(Xl， hfつ
，(X2， M*)，…， (Xm， h長*)>} so thatM*(i= 1，… ， m) corresponds to thek*th highestR(xω T h e fused possibi1ity grade of the-128ー Kagawa UnzversiかEwnomicReview ithsample x， is as follows : h{=
L
:
wk*M*(i = 1，… ， m)， J毎本ニ1.. .P 386 は り where the weightWk*is determined by O W A operators [20].. Using the method introduced in Section 3.，2.the new dual possibility distributions， denoted asXt~<
(α" Dtu)e， (αi， Dt山>， can be obtained from (Xi， h{)(i= 1，… ，m).. It is obvious that the formula仰)is idempotent， that is， if all experts have the same knowledge， the fused possibility grades should be the original ones， and symmetric， that is， initial indexing of the arguments doesn't matter..
5
.
The Portfolio Selection sased on Fused Possibility DistributionsPortfolio selection problems based on possibility theory have been studied in [8， 15， 16].. Different from the probability model， such as Marl王owitz'smodel， where optimal portfolio is selected based on the statis
-tic characteristic of the past security data， the possibility model select the optimal one based on the past security data and experts' judgment on those data， where the possibility distribution is used to characterize the experts' knowledge.. N ow let us consider portfolio selection problem with multiple experts The data is given as {< (X1， h，)l (X2， hD，…， (Xm， h~) >，…，< (X1， hf)， (X2，
M
)
，…， (Xm， h品)>} where Xk = [Xk1，…， Xkn]t is a vector of returns ofn securitiesSj(i= 1，… ， n) at thekth period， h~ is an associat -ed possibility grade given by theithexpert to reftect his judgment on the possibility degree that such returns ofn securities will appear in the future， and s is the number of experts. Using the method introduced in Section 2. 3， the fused possibility distribution Xf'
"
'
-
'
<
(
αi， Dfu)e， (αi， D/l)e> is387 Decision Analysis Based on Possibility Theory -129-from multiple experts.. The portfolio return can be written as z = rtx = ~ r品， ;=1・ ，，n (48) where r;denotes the proportion of the total investment funds devoted to the security
S
;
and め isits returu. Because the return vectorx is governed by the dual possibility distribu -tionXf"-'くα(f，Dfu)e， (αf， Dfl)e>， using Theorem 2 the dual distributionsof a possibility portfolio returnZ， denoted asZ"-'く7[z.(z)，ぬ，(z)>， are obtained as follows : Rん(z)= exp{一(z-rtaf)2(rtD向r)一1}= (rtαf， rtDfur)e， 似の ぬ
μ)
= exp{一(z-rtaf)2(rtDμr)ー1}= (戸時，rtDμr )e， (50) where 戸時 isthe center value， rtD舟，rand rtDflr are the spreads of a possibility portfolio return Z based on fused upper and lower possibility distributions. Considering the fused dual possibility distributions， the following two quadratic programming problems to minimize the spread of possibility portfolio return are given where the spread of possibility portfolio return is regarded as the measure of risk ロ1m r min rtDfur ア Sゅt.rtαf C， n zrz=l， ri二三 0， rtDflr Sれt.rtαf C， n呂
r
z
= 1， ri二三 0， ( 51) (52)Kagawa University Economic Review
-13(}ー 388
where c is the expected center value of a possibility portfolio return which should comply with the constraint min ai::;:c ::;: maxαi to guarantee the
i=l， '，n i:=l.，n
existence of the solution is (5)1and (52). BecauseDfu and Dfl are positive definite matrices， 5(1)and (52) are convex programming problems.
Consider the following optimization problem where short share is allowed in portfolio selection problem. min rtDr γ (53) s引t r tα C，
呂
n=
1，where D is either DfU or Dfl. The optimal solution r* can be obtained by minimizing the following Lagrangian function， L(r， I，'1 A2)= rtDr+A
μ-rta)+
ん(1一戸1)， (54) where 1 = [1，… ，1]t L(r， A1， A2) is a convex function because of D >0 The necessary and su伍cientconditions for optimality of同 are oLjor= 0， OLjOA1= 0， OLjOA2= 0， (55) where can be explicitly written as 2Dr-A1α A21=
0， )) ) Runt-oo ﹁ 吋 A M E d ﹁ 吋 d ( ( c-rtα= 0， 1-rt1 = O. From (56)， we have rt = 1j2(A1atD-1+A21tD-1) (59) Substituting (59) into (57) and (58) leads to the following equations.. c = 1j2(A1atD-1α+A21tD-1α)， 側 1 = 1j2(A1atD-11+
A21 tD-11)“ (61)389 Decision Analysis Based on Possibility Theory
-131-For simplicity， we let
α= 1/2atD一α，1 β=1/21tD-1α， y = 1/21tD-1L (62)
It should be noted that仏 βandy are constant values. Thus， (60) and (61) can
be rewritten as
α'/h+sA2 = c， (63)
ßÀ 1十~=1 ~
Assuming thate
=
αy-s2 is not zero， we can solve the equations(63)and (64) to obtainA1 and A2 as follows : ん=(cy-s)/e， A2 = (α-cβ)
/
e
(65) (66) Substituting (65) and (66) into (59) leads to r*t= (cr一β)/2eatD-1+(α-cs) /2erD-1 Thus， (67) r*=
((γ!2e)D-1α一(s/2e)D-11)c+ (α/2e)D-1j一(s/2e)D-1α)
=
bc+d， (68) where b = (y/2e)D-1α一(s/2e)D-11， (69) d = α(/2e)D-1j -(s/2e)D-1α (70) Because b and d are constant vectors， it follows from (68) that the optimal solutionr集 isa linear function of the given center c Considering thatr*tDr* is the smal1est spread of the portfolio return denoted as r， we have r = r*tDr* = (btc+dt)D(bc十d)= c2btDb+2cbtDd+dtDd(7I) SincebtDb， btDd and dtDd are constants denoted ast1， t2and，ら respec -tively， (7I)can be simply written as fol1ows r = t1C2
+
t2C十t3， (72) which means that the spread ' i['s a quadratic function of the given center c .. Theorem 5. The spread of the possibility portfolio return based on the fused-132- Kagawa University Ewnomic Review 390 lower possibility distribution is not larger than the one based on the fused upper possibility distribution
Proof. Suppose that the optimal solutions obtained from the problems (5]) and (52)are denoted asr~ and ri， respectively， with considering the same center value十 Accordingto the feature of the upper and lower possibility distributions， i..e川Dtu-Dfl 主主 0， the following inequality hold.. r~t Dtu付与 r~t Dflr~ Because ri is the optimal solution of(52)， we have r~t Dtlr~ 主主 ritDt川 ¥ As a result， r~t Dtur~ 主主 rit Dnrl¥ which proves the theorem
。
3) ( 74) (75)口
The nondominated solutions with considering two objective functions， ie， the spread and the center of a possibility portfolio in the possibility portfolio selection models (5])and (52)can form two efficient frontiers
The portfolio selection models based on necessity measures is given as follows: 立lax y Nes((rtx)ミ c) n s.t
呂
ri= 1， r注0， (76) where x is governed by dual fused possibility distributionX ~<
(αt， Dtu)e， ( αt， Dfl)e> Its dual approach is given as follows 立lax C 仰) s.t Nes(rtx注 c)注 α， n呂
ri= 1， r 二三o
.
391 Decision Analysis Based on Possibility Theory 133-h c q y Fig3 Explanation ofNes(r'x注 c ) 注α Considering the possibility distribution ofx， Le..X ~
<
(αf， Dtu)e， (αf， D/l)e>， it leads to N白(rtxミ c)二三 α~rtaf-J-lri(I α)rtDr ~と C， (78)where αis a necessity level given by decision-makers and D is eitherDfU orDfl A graphic explanation is give泊 inFig..3， which shows that the
feasible region forc is [0， q] where q = rtαf-
Ff
五日一 α)rtDr Itfol -lows from (78)that maximizing the parameter c leads tompx rV-J-fn(1一α)r'DAr. Thus， the problem (77)can be rewritten as
max rtα-v'-ln(l一α)r'Dr n s..t
呂
n= 1， 約二とO (79)。
。
134ー Kagawa Universi'ty Ewnomi:ιRevieω 392 It should be noted that the model (80) is a convex programming problem.
6
.
N
umerical Example In order to show the above-proposed approaches， a numerical example for the stock market analysis is given. Table 2 lists the security data with associated possibility grades for stock return prediction given by five experts.. The dual possibility distrぬutionsfrom the expert 1 to 5， denoted asXl~<( α1， DU1)e， (α1， Dl山>， X2~ く (α2， DU2)e， (α21， D12)e>， X3~< (α3， Du3)e， (α31， DI3)e>， X4~<( α4， Duふ， (α4， D心
e>and X5~< (α5， Du5)e， (α5， DI5)e>， were obtained as follows“Table 1 Security data with experts' knowledge
Years Expert 1 Expert 2 Expert 3 Expert 4 Expert 5 Sec 1 Sec 2 Sec 3 Sec 4 1977(1)
o
881 0.805 0.192 0..192 0..21 -0..305 -0.173 -0..318 -0..477 1978(2) 0.279 0.273 0..901o
621 0.69o
513 0..098 0..285。
..714 1979(3) 0.52 0.602 0.54 0.685o
672 0.055 0..2o
047 0.165 1980(4)o
623 0.671 0.517 0..542 0.557 -0..126o
03 0.104 -0..043 1981(5)o
811 0.877 0.312 0..298 0..277 -0..28 -0.183 -0.17l -0..277 1982(6) 0.522 0..565 0..623 0.58 0..536 -0003 0..067 -0..039 0.476 1983(7) 0.377 04
.
31 0.676 0..918 0.782 0.428 0..3 0..149 0.225 1984(8) 0..46o
452 0.698 0..717 0..726 0..192o
103o
26 0..29 1985(9) 0..348。
374 0..716 0..809 0..877 0..446 0.216 04
.
19 0..216 1986UO)。
..736 0.797 0.371 0.414o
41 -0088 -0..046 -0..078 -0272 1987UJ) 0.598 0.667 0.556 0..622 0.575 -0..127 -0..07l 0..169 1988U2) 0.588。
673 0..54 0..628 0..582 0..015 0..056 -0.035 0..107 1989(13) 0.475。
.
4
84 0709 0.753 0..696 0..305 0..038 0.133 0..321 1990Ul4 0..415 0.434o
535 0..468 0.615 -0.096 0..089 0..732 0.305 1991U5)o
561。
582 0..581 0.649 0..639 0..016o
09o
021 0.195 1992U6) 0.443o
503o
669 0..681 0..671 0..128 0.083o
131 0..39 1993U司 0..611 0.689 0..482 0..582 0.522 -0.01 0..035 0..006 -0.072 1994U8) 0.222 0..223 0..661 0.405 0..546 0..154 0..176 0..908 0..715393 Decision Analysis Based on Possibility Theory 135 α1
=
[-0..305， -0173， -0318， -0..477]t， 321..045 105..691 -444..331 93..502 105..691 35..305 -146..258 30..933 DUl=
I
-444..331 -146..258 618..334 -128..763 93..502 30..933 -128..763 28..420 320..616 105..648 -444..659 93“
464 105..648 34..876 -146..402 30“
880 Dll=
I
-444..659 -146..402 618..023 -128..961 93..464 30..880 -128..961 27..670 α2=
[-028， -0..183， -0..171， -0 277]t， 296..981 131..778 -436“
389 15..250 131..778 59..758 -194..312 6..845 DU2=
I
-436..389 -194..312 644..674 -21.834 15..250 6..845 -21.834 1.760 1..078 0..532 -0..913“
。
379。
..532 0..310 -0..346 0..254 D12=I
-0..913 -0..346 1..965 0..260。
..379。
..254 0..260 0..523 α3=
[0..513， 0..098， 0.285，o
714]t， 82..803 -259..691 59..369 -63..082 -259..691 825..545 -188..827 202..048 DU3=
I
59“
369 -188..827 44凶471 -46..040 -63..082 202..048 -46..040 50..606。
ω695 0..111 -0“
255 O刷149 0..111。
..133。
..154 -0..085 Dl3=
I
-0..255 O副154 1.144 0..073 0..149 -0..085。
..073 0..648-136- Kagaωa Univer.sify EじonomicReview 394 α4
=
[0428， 0..3， 0..149，o
225]t， 0..728 0..385 0..053 0..385 0..379 0..073 一0..162 DU4=
I
-0..053 0..073 1伽107 -0..065 0..065 -0..162 -0..065 l“086 0..405 O “180。
..032 0..180 0..246 0..164 -0..106 D14= 0..032 0..164。
..698 -0..440 0..099 -0..106 -0“440 0..736 α5=
[0446， 0216， 0419，o
216]t， 0..950 0..248 0..129 0..248。
..339 0..095 -0..010 0..129 0..095 0..922 0..109 0..008 -0..010 0..109 0..921 0..595 0..177 0..118 0..177 0..146 0..096 0..118 O“096。
738 -0..140 0..275 O “076 -0..140 0..329 The obtained consistency index matrices based on upper and lower possibility distributions were listed in Table 2 and Table 3， respectively.. The binary matrixQ* was obtained in Table 4 withゐ =0..85and ふ =0.68 From Table 4 it was known that the e伍cientblock was Ue = {X3'"<
(α1，Du3)e， (α3， Dde>， X4"'<(α4， DU4)e， (α4， Dl山>， X5"'< (α5， DU5)e， (α5， DI5)e>ト The reliability degrees of R(X3)， R(X4) and R(X5) were obtained as R(X3)
=
3..156， R(X4)=
3..378and R(X5)=
3414.. The weight coe伍cientsofM
，品 andh% (k= 1，ー， 18)were set as1/6， 1/3and 1/2， respectively. Using the weighted possibility degrees， the fused possibi1ity distribution Xf" ，<
(αj， Dru)e， (α/， Dll)e> were obtained as395 D巴cisionAna!ysis Based on Possibi!ity Theory α1 = [0..446， 0..216， 0..419， 0..216]t， 0..988 0..275 0..001 0..275 0..277
。
..075 0..001o
075 1..134 -0..070。
..129。
..020 -0..070 0..918。
“
710 0..197 -0..042 0..355 0..197 0..168 0..061 0..095 -0..042 0..061 0..993 -0..256 0..355 0..095 -0..256 0..369 Tab!e 2 Consistency index matrix based on upper possibi!ity distributions x1 x2 x3 x4 x5 x1 1o
993 0..503 0..612 0..563 x2 0..993 1 0..545 0..651 0.603 x3 0..503 0..545 l 0.879 。引856 x4 0.612 0.651 市。879 1 0.939 x5 ~空3LQ
型3
o
856 0.939 1 Tab!e 3 Consistency index matrix based on !ower possibi!ity distributions x1 x2 x3 x4 x5 x1 1。
..92o
263o
39心
。
383 x2 0.92 1 0..35伽
。
506o
49 x3。
..263 0.35 1 0.681、
。
74 x4 0..39 0..506 0..681 1 0..879 x5 0..383 0..49 0.74 0.879 1 Tab!e 4 The binarγmatrix of consistency index x1 x2 x3 x4 x5 x1 1 1。
。
。
x2 1 l。
。
。
x3。
。
1 1 1 x4。
。
l 1 1 x5。
。
l 1 l -137二一396 Kagawa University Economic Revieω
-138ー
Using formulas (51)and (52)， the possibility portfolio frontiers based on The port -upper and lower possibility distributions were shown in Fig 4
folios based on fused upper and lower possibility distributions with c = 0..3 was shown in Fig 5，
o
5 0，4 0，，3 0，，2 匂 国 心 ﹄ 己 的 0，1。
0..4 0，35 center 0，，3 0，25 Portfolio frontiers based on upper and lower possibility distributions 10ド日% 4 35.6% 15，9% Fig4 Portfolio based on fused lower distribution Portfolio based on fused upper distribution Fig5 Portfolios based on fused upper and lower possibility distributions with c=0，3 Concluding' Remarks7
.
In this paper， from upper and lower directions the upper and lower possibility distributions are used to approximate the given possibility The upper possibil -grades， which is regarded as the expert's knowledge，
397 Decision Analysis Based on Possibility Theory -139-ー
ity distribution ref1ects the optimistic viewpoint of the expert and the lower possibility distribution ref1ects pessimistic one. Di丘erentfrom probability distributions for ref1ecting the statistic characteristic of data， possibility distributions are used to characterize the human knowledge so that multi-source knowledge from multiple experts can be represented by a set of exponential possibility distributioss. Based on the consistency index defined by possibility measure of each pair of possibility distributions， the fusion model is proposed to integrate the multiple possibility distributions into a new one representing a refined knowledge“ As an app1ication， a portfolio selection problem with multiple experts is considered Because the fused possibility distribution has higher credibility than the single one， it is natural that the more reasonable portfolio can be obtained.
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[9 J Z Pawlak， Rough Sets (Kluwer Academic Publishers， Netherlands， 1991)
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