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Properties of the fuzzy expected value and the fuzzy expected interval
Sets and Systems 26 (1988) 373--385 North-Holland
373
P ~ O P E ~ T ~ E S O F T H E F U Z Z Y EXPEC-WED V A L U E M F U Z Z Y EXPECTED I N T E R V A L ~ Moti SCHNEIDER and Abraham K ~ ' D E L Deparlment of Computer Science and the Institu~ for Expert Systen~ and ~bo~s, Flork~ State Umvers~, Tailahassee, ~ 32306, USA Received April 1986 R e v ~ d July 1986 The evaluation of the fuzzy expected value (~'~V) r©qui'tesa complete knowledge about the domain of the evaluation, and the distribution of the population in that domain. Since it is not always possible to assmne a complete knowledge about the domain, it is necessary to find some relaxations to the ~stdctions involving the evocation of ~ V . In this paper some solutiom ~o tb~ problem are proposed via the concept of the fuzzy expected interval (FEI).
Keywords: Fuzzy expected value, Characteristic functions, Fuzzy expected interval.
1. I a ~ c ~ c f l ~ a
In order to compute the fuzzy expected value ( ~ V ) , as well as any other measure of typicality, it is necessary to know how the population is distributed and what is the grade of membership of each group. In other words the fuzzy expected value may not be applicable when partially unknown or incomplete information is given. In this paper we present several methods to overcome this problem. First we discuss cases in which the fuzzy expected value will not be affected when some incomplete data is utilized. We then proceed to show how to appro~mate the fuzzy expected value in a fuzzy environment by introdudng the fuzzy expected interval. The fuzzy expected interval is found to be useful in many cases where the fuzzy expected value cannot be found. Moreover it wit] be shown that in some cases the fuzzy expected interval can be reduced to a fuzzy expected value even when partially fuzzy information is availab|e~
2. Fu~y Elpeded V~ue Let XA be a B-measurable function such that XA~ [0, I]. The Fuzzy Expected Value (FEV) of XA over the set A, with respect to the fuzzy measure ~(.), is defined as su
lj (rain[r,
* This work is partially supported by NSF grant IST8~5953. 0165-~114/88/$3.50 © 1988, Elsevier S~iencePublishers B.V. (North-Holland)
(1)
374
M. $chnekler, A. Kandel fA(T)
l'k
fA(T)
HI---
0
H
T
1
Fig. 1. The evaluation of FEV(gA).
where
{x [
r}. Now,
T} =fA(r) is a function of the
threshold T [t]. The function ~ maps ~ into the interval [0,1]. The actual calculation of FEV(xA) consists of finding the intersection of the curves T--f.4(T), which will be at a value T = H, so that FEV(gA)ffi H ~ [0, 1] [2]. Figure I depicts this procedure while the following exsmple illustrates how the FEV can be computed. ~p~ L Using the base variable 'hourly wages', assume a given population and a given subjective compatibility curve such that I person earns $3.00--~X = 0 . ~ ,
3 persons 4 persons 2 persons 2 persons
earn earn earn earn
$ 4 . 0 0 ~ X - 0.50, $4.20~X - 0.55, $4.50~ X = 0.60, $10.004 X "- 1.00.
As can be seen we have 5 different thresholds (0.40, 0.50, 0.55, 0.60, 1.00). The first step in the process is to check how many persons are above each threshold (in pex~ntage terms). As can be seen, 12 persons are above or equal to 0.4, 11 persons are above or equal to 0.5, 8 persons are above or equal to 0.55, 4 persons are above or equal to 0.6 ~md 2 persons are above or equal to 1.00. Pairing these data and rearranging by increasing order (of the measure of belief), we obtain the following five [7", ~] pairs (see Eq. (1)): (0.40, 1.00),
(0.50, 0.91),
(0.55, 0.66),
(0.60, 0.33),
(1.00, 0.16).
Now, the minimum value of each pair is: min(0.~, 1.~) = 0.40, min(0.~, 0.33) -- 0.33,
rain(0.50, 0.91) ffi 0.50, rain(1.00, 0.16) - 0.16,
rain(0.55, 0.66) -- 0.55,
.Fuzzy~ z e d v ~ and fuzzyetpec~dintenJal
375
and, therefore, following ~ . (I), the FEV, which is the maximum of all these m i n ~ a , is: max(0.40, 0.50, 0.55, 0.33, 0.16) ffi 0.55.
Thus the FEV is 0.55. From this result we can obtain that the fuzzy expected hourly wage is $4.20. Another approach to find the fuzzy expected value is by arranging all w~lues in increasing order and to find the median. The median is the FEV. Thus
the median is 11/20-0.55 and so is the ~ V . The following theorem proves that the median of the seqeence is the I ~ V .
~ e o ~ m 1. The medi,~n of the set of 2n - 1 numbers, represenffng aU the X~s Cn numbers) and all but t~,efirst ~'s (n - 1 numbers), where 0 <~X~ < X2 < " " < Xn <~ 1 for finite n, arranged'l~n order of magni~de, is the FEV [3]. Establishing the fa~L that the FEV is the median of an ordered list is' very helpful in the developr~lent of new prope~ies about the F]EV. But first we discuss the issue of the characteristic function.
3. The e b M e t e ~ e
~e~m
One of the most important issues involving FEV is choosing the appropriate characteristic function which is used for mapping from the actual values to the interval [0, I]. Choosing the wrong characteristic function may result in wzong interpretation of the mapping, and thus make the whole evaluation of the FEV meaningless. There are two main goals to consider: I. The characteristic function must be monotonic to avoid ambiguity. 2. The range of the characteristic function must be sufficient to cover aU possible cases (and only these cases). ]Example 2 demonstrates how to construct a meaningful characteristic function for a particular population. E _r~m_~e 2. Suppose we have the following characteristic function for the variable OLD: X(x)
~x/lO0 h
if x ~ 100, fix > 100.
From the characteristic function described above it is clear that the function covers all ages and that the mappings are one to one. Thus, it is very important to construct a characteristic function such that the mapping will be one to one, and this function must cover only the relevant cases. By doing so, we can guarantee logical mappin~ from the actual nun~bers to the
M. Schneider, A. Eandd
376
corresponding X's. Next we discuss the concept of the fuzzy expected interval
4. ~ t e r v l
of ~ V
In general, whenever there is a population divided into n groups with n distinct X's we have
different arrangements of the set, and possible 2n - 1 different FEV's. Example 3 illustrates these points. E x ~ p t e 3. Let X be a pop~ation sucE that x people are of age = (compatib|e value X~), y people are of age p (compatible value X2), z people are of age 7 (compatible value X3), where
Z _y+Z
x+y+zfX,
~4
O<~x~
Since n = 3, there are 10 different arrangements as given below. Z2
Zz
z X
z X
y+z X
Xz
X2
3 Xl
~,
Xz
X3
4 Zl
~
X2
g~
,/k
1
Z~
2
z
z
y+z X
Z3 ,y +z
y+z
-~"
X3
Z~
y+z X
Zs
Z~
X2
E~
Z~
X2
z ~
8 x~
X2
~
9 Xl
~
z
y+z
5 6
7
z X Z
z
tO ~
y+
Z~
z
T
y+z
-~-
y+z X
X3
Z
X x3
y+z
X2
X3
x2
X3
It can be seen that xow 1 has FEV = X3, tow 2 has F E V = Zl, iows 3 through 6
F~¢zzyexpec~edvalue and fuzzy expectedintervM
377
have FEV=X2, rows 7 and 8 have FEV = z / X and rows 9 and 10 have F E V -- (y -~ z)]X. Thus the possible number of FEVs is 5, as was stated at the beginning of the section. Now suppose that x people above are of age o? with value X~, such that X~ median or 1" and 1,. < m e d ~ n and l* ~ {il, . . . , 12~-1} will not change the FEV.
~f. If both l~ and I* are greater than I. then only the upper n - 1 numbers in L may have different arrangements, and thus I. is still the median. In case both !*~ and 1,,, are less than In then only the lower n - 1 elements in L may have different arrangments, and In will still be the median. Since the median remains the same, so is the FEV. [:] "l%eerem 3. Let L be an ordered set of all X's and ~'s and l~ be the med~n. Then changing 1., to l* ~ {Ira - e, In, + e}, where 1" is some Xj such that l~ - e > Ira-1 and l,,, + e < Ira+l, will not change FEV.
~f. Let Theorem 2 both ~ and change the
a~= l,n - e and ~ = l,n + e. Since ~ is distinct then according to it will not change FEV. We use the same argument for ~5. How, since fl will not change FEV, then any number in the interval [~, IS] will not FEV. El
If the change of 2~i to X* results in X* =X j, then we combine the two populations and reduce n by 1. The importatJce of Theorem 3 is that it allows us to manipulate imcomplete (or fuzzy) information. Example 4 illustrates this point. E ~ m p [ e 4. Suppose we have population A and the following distribution: Pl
Xl
P2
X2
*
m
o
°
Pi
Xi
o
Q
*
o •
Pj
X~
*
•
i
Q
m
m
Pn Xn where A --Pl 4"P2 4--. ° +p~, aIt~ Xi is the FEV. Now substitute pj and £j with pj
378
M, ~hne/der, A./~'~u~el
~ d move or less X~. We ~ h~e~p~et, ~o~¢ [ ~ - e, ~ + e]. Now the new distribution is: P; P2
X~ g2
P~
Z+
•
P,,
~ ~jf
~,
According to Theorem 3 the I ~ V wil not change. Thus it is possible to use some fuzzy informatio~ and still compute the FEV. Moreover, using incomplete hu~ommtio~ will not cha~ge FEV. In conclusion we can state the following corollary: Cen~. ¢ ~ g m g all X's (except the ~ equal to the FEV) to/ntewa/.~ such that ed[ intewols ave disjoint and al~ ~'s ave no~ in d~.se intewals, w ~ not change F~V. ~ e e f o L e t e ] = ~ j - e , Xj+e], j - l , 2 , . . . , 2 n - l , sad let fl]e~], ] I, 2 , . . . , 2n - 1. Then according to Theorem 3, we can change fll to fl~' • ~ and F~V ~ . not be affected. Changing any ~j to fl~ ¢ ~ (such that n :~]) will not change FEV.
The concept of the ~ expected interval was developed in order to cope with incomplete (or fuzzy) information. The ~ I must be interrelated with ~ V in such a way that the result of the evaluation will be either ~ V or FEI but in both e.~es it must have a meaningful mapping back to the o r i ~ a l numbers. In this section we consider the case where the X's may or may not be represented in intervals, however, these intervals must be disjoint. In addition, we allow the/Fs to fall into the intervals of the X°s. Consider the following example. Emmple $, Let A be a population such that x people are within the age range of ~ to ~2 (computable values Xt to X2), y people are within the age. tango of ~ to f12 (computable values X3 to X4), where A = x + y and X~< X~< X3 < X4. Now we have the foHowhlg arrangements:
I
x2], L 3- x,,]
Fuzzy expec~d v~l,ueaml fuzzy expeaed t~ceruM
3
l'l
4 Lxt -" X2], L~3- g4] when y falls into one of the intervals. First 3 cases present no problems, s ~ we can use Theorem 1 and make the median the FEV (case 2) or FEI (eases 1, 3). But what about case 4? Since in e.~l~4 th¢l¢ is no median we have to seek a differem solution. O~xecan see that if y/A falls inside the int©rval L~, X2] i~ is reasonable to assume that the FEX will be in that in,real. The same argument may be applied to the case where ylA fails into the interval IX3, X4]. To make the decision process systematic we can define the following role (based on Theorems 2, 3 and the corollm'y): R ~ 1. Whenever some ~ falls inside a given interval, delete the $, and duplicate th© interval. In conclusion, we can say that using incomplete information doesn't have to d ~ ' t our decisions. By providing a complete characteristic function and some information about the population (or the domain in which we have to make a decision), the F~V or the ~lEI can still be computed. Consider the next example. 6. Let the characteristicfunction of the variable O L D be:
x(x) =
/1oo
ifx <0, if x ~<100, if x > 100,
and rite dis~bution of the population is 10 people are of age range 10--20, 25 people are of age 30, 15 people are of age 40, 30 people are of age range 45-55, 20 people are of age range 60-70. Now, translating the actual numbers to ~'s and g's we get
1,00 0.90 0.65 0,50 0.20
0.1-0.2 0.30 0.40 0.45-0.55 0.6--0.7
M. Schneider, A, Kandel
380
Ushg ~
'"
[oa-0.2],0.3,
[0.6-o.7],[0.6-0.7],0.90
and the median is [0.45-0.55].Thus the FEI is [0.45-0.55].Going back to the originalnumbers we conclude that the fuzzyexpected age ~ between 45 and 55. It is easy to see that if we had 0.55 instead of the range [0.45-0.55] then the median would be 0,5 and the fuzzy expected age would be 50. Now we can present the theorem about the relation between FEV and FEI.
~ e o r e m 4. Let L ffi {11,. • •, l , , . . . , 12,-~} be an ordered list of I~'s and X's such that any X~ can be represented by either a single valued number or by an interval. Then FEV ¢ FEI. Proof. Let FEI = { s ~ , . . . , s~} be ~he median of L. Then FEI must contain the FEV, since any selected s~ ~ FEI will remain the median, and thus s~ -- l,, -FEV ¢ FEI. D Next we consider the case where the intervals may overlap. 6. ~
the FEV ar ~he FEI in o v e ~ p ~
~tervgq
The problem with intervals which are overlapping is how to deal with the case where a certain/~ may fall within the intersection of two or more intervals. In this case Rule 1 will not be applicable since Rule 1 is appropriate only for cases in which the intervals are mutually exclusive. One possible solution to this problem is to break the intervals in such a way that the intervals will become disjoint. Example 7 illustrates this point. ~~e
7. Let A be a population and let the distribution of A be: X people are within the age range of ~r1-¢2, Y people are within the age range of ~ 3 - ~ ,
such that ~1 < ~3 < c¢2< ~4 and A = X + Y. The regular solution will be to find all possible arrangements. Now we can break the population in the following way: xl people are within the age range of ah and c¢3 - e, X - x t + Y - y a people are within the age range of at3 and o:2 - e, Yl people are within the age range of c~2 and ~4. such that X = x~ + x2, Y = yl + y~. and ~:he intervals are disjoint. Assume that a relevant characteristic function generates the following mapping: ~:"> Xl,
~3 - e--->X2,
~:"> X3,
Fuzzy expected value and fuzzy expec~d ~ r v a l
381
Now we ¢a~ pair the data:
X - xt + Y - y~ + yl A Yl
L% - X~]
L~-~,]
We can see the problem. Since we have no information about x~ and y~ we can not find the FEV nor the FFI.
It is clear that we have to find a differem Way to get around this problem. Thus we have to establish a rule to handle cases where a given ~ may fall in more than one interval. (The following rule is established emph~cally, the justification will follow in Theorems 5 through I0.) Rule 2. Whenever a certain/~ falls in more than one interval, choose the interval such that the ~ falls as close to its nfiddle as possible. After choosing the hlterva~, delete the ~ and duplicate the interval T ~ o u g h the use of Rule 2 we can now solve the problem in Example 7 by creating the following arrangements (assuming the mapping: ~--> X~):
-Y L~-- x~], L~3 - x,], A' Y
[X~ - X2], [Z~ - X2], [Za - X4] if Y is closer to the rlliddle of the f ~ t interval, .Y [Z1 - X2], L~3 - x4], L~a - x4] if ~ is closer to the middle of the second interval. We can formalize the rules by writing the next six theorems.
Def~liom 1. Let S and R be two intervals such that $ = {sl, .... ,s.} and R = { r ~ , . . . , r,,,}, then: 1. MAX(S,R) ffi $ ff for every s~ ¢ S there is rj ¢ R such that s~ > rj. 2. t~rN($, R) = S ff for every s~ • S there is r] ~ R such that ss < rj. Note. We assume that each interval may contain a single element,
" £ h ~ m S° Let $ and R be two iaterva~ such taut (a) S = { s ~ , . . . , s,}, (b) R = { r ~ , . . . , rm}, and (c) R n S = ~. Then
~(S,R)=
{~ ifr,>s.. if s~ > r,,,.
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382
~f~
M. Schnader, A, Kandd
Let s~ ~. r~.. Then . . b~sed ~n D e f i ~ f l ~ lqL~, ~.,..,_x($,R) - S,. []
~e~m R ffi { r ~ , .
6. Let $ and ~ be avo intervals such that (a) $ ffi { s ~ , . . . , s,}, (b) . . , r m } , and (c) R N $ = ~ . Then {~ ~ ( $ , R) ffi
ifrm
(4)
~ o e f . Let r,,, < s~. Then based on Definition 1(2), urn(S, R) ffi R. Theorem 7. Let S and R be two intervals such that (a) S ffi { s ~ , . . . , s , } , (b)
Rf{rt,...,r,,,},
(c) R N S ~ ,
~L.x(& R) ffi {R
(d) $ ¢ R a n d R ~ $ ,
a n d ( e ) s n > r ~ . Then
ifrm>s,, if s~ > rm.
(5)
Proof. Let rm > s,,. Then based on Definition 1(1), ~o,x(S, R) = R.
r3
"~eorem 8. Lea S and R be two imervds such that (a) S ffi { s l , . . . ~ an}, (b)
R f f i { r l , . . . , r ~ } , (c) R N S ~ , ~ ( $ , R ) _ _ {R
(d)$¢RandR¢$,
o~d (e) s, >r~. Then
ifr~
(6)
if s. < r..
Proof. Let r~ < s,. Then based on Definition 1(2), ~L~($, R) ffi R.
D
To make the theory complete we need the foUowing definition. De~,1eon 2. Let $ end R be two intervals such that $ = { s l , . . . , s,} and R ffi ( r ~ , . . . , rm} and R _=$. Then there is an interval T such that: 1. uAx($, R, T) = 7", ff for every t~ ¢ T t~ere is sj e S such that tj ~ sj and for every ta ¢ T there is vk E R such that ta ~ r~. 2. ~ ( $ , R, T) = T, if for every t~ ¢ T the,re is sj e $ such that t~<~sj and for every tte T there is rk ¢ R such that tt ~ rk. T h e e r e l 9. Let $ and R be two intervals' such that 3 ffi {sl, . . . , s,} and
R ffi {v~, . . . , vm} and R =_$. Then
MAX(S,R) {rl,..., s.). =
(7)
P ~ f . Let T be an ordered set such that T - { r x , . . . , s,}. Then based on Definit/on 2(1), MAx(S, R, T) -- 7". 0
Theorem 10, Let S and R be two intervals such that S ffi{s~,..., sn} and R - { r ~ , . . . , rm} and R =_S. Then MIN(S, R ) =
. . . , rm}.
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Fuzzy expecteduMueand fuzzy expectedintevual D e f ~ o n 2(2), Mr~q($,R, T) = Z
383
El
Defe~lieu 3. Let ¢ and tff be intervals. Then we say that ¢ i~ higher than ~ ff the upper bound of ¢ is greater than the upper bound of ~. KvJ~ple 8. Using F,q. ( l ) and the data from Example 6, we can form the following pairs: /A
Z
1.00 0.75 0.65 0.50 0.20
0.1-0.2 0.30 OAO 0.45-0.55 0.6-0.7
Now, according to Eq. 1 we have to find the M=N for each pair:
~N(l.oo, [o.l-o.21) = [o.l-o.2]
(Eqo 4),
MIN(0.?5, 0.3) = 0.3, ~uN(0.65, 0.4) = 0.4,
~(0.5, [0.45-0.55]) = [0,45-0.50] ~uN(0.2, [0.6-0.7]) --- 0.2 (Eq. 4).
(F I. 8),
The next step is to find the MA× over all the ~Ns:
~x([0.1-0.2], 0.3, 0.4, [0.45=0.50], 0.2)-- [0.45-0.50]. The following algorithm summarizes the steps involved in the computation of the FEV or the FEI: 1. Find eli the ~'s and X's using Eq. (1) and the characteristic function, respectively. 2. Find the win of each pair using Eqs. (4), (6) or (8). 3. Order the results above into an ordered list L based on Definition 3. 4. Find the wax of L utilizing Eqs. (3), (5) or (7). The following example demonstrates that the tools introduced in this paper allow the possibility of computing FEV in spite of the fact that all Z's are given as intervals. Ex!mp|e 9. Let the characteristic function of the varieble OLD be:
x(x) =
/loo
if x < O, if x ~<100, if x > 100,
384
~
M. Sc3me~,v, A./Candd
~r~ ~ b ' ~ o n
ef the population be:
30 people 20 people 30 people 20 people
are are are are
of the of the of the of the
age age age age
ranging from ranging from ranging from ranging from
10 to 40 to 50 to 80 to
20, 50, 70, 90.
According to step 1 of the algorithm described above, the following pairs are computed: 1.00 0.70 0.50 0.20
0.1-0.2 0.4--0.5 0.5-0.7 0.8-0.9
Now, the ~NS Of the pairs are (based on step 2 in the algorithm): ~s(1.00, [0.1-0.2]) = [0.1-0.2] ~uN(0.70, [0.4--0.5]) = [0.4-0.5] u (o.5o,
- o.5
(Eq. 4), (Eq. 4), g),
~ ( 0 . 2 0 , [0.8-0.9]) = 0.2 (Eq. 4). Thus the ordered Hst of the Mn~scomputed above is:
L-- {[0.1-0.2], 0.2, [0.4-0.5], 0.5} Now, according to the algorithm the maximum of L is MAX([0.I-0.2], 0.2, [0.4--0.5], 0.5) = 0.5 (Eq. 7). ?. C e a e ~ l e n In this paper we have ir:~oduced some issues related to the concept of the fcz~ expected value and the theory of typicality, it is shown that by adding the concept of the fuzzy expected interval we can solve some interesting problems involved in these issues. The use of the fuzzy expected interval is found to be very useful when the information regarding the domain and the distribution of the popu|ation is partially unknown or incomplete. Moreover, in some cases we are able to reduce the s~e of the interval and map that interval into a single valued number when going back from the characteristic function to the original data. The fact that we can compile incomplete data is quite usefu! in dealing with fuzzy variables and building fuzzy expert systems [4--9].
Rdem [1] A. ~nde! and W.L Byatt, Fuzzy sets, ~uzzy algebra, and fuzzy stads~cs, Prcc. IEEE 66 (i2)
Fuzzy expected value and fuzzy expectedinterval
385
I21 A. KandeL Fuzzy Te~eniques in Pattern Recognition (Wfley-Interscieuce, New, York, 1~2). [3] A. ~tndei, Fuzzy Mathemaucal Techniques v~th App|ications (Addi~n Wesley, Rea~ng, MA, 1986). [4] C.V. Negoita, Expert System and Fuzzy Systems (Benjarain/Cummln~, New York, 1985). [5] S.M. Weiss and C.A. Kulikowsld, Designing Expert Systems (Rowman and A~anheld, Totowa, NJ, 1984). [6] F. Hayes-Roth, D.A. Waterman and D.B. Lenat, Building Expert Systems (Addison-Wesley,
Reading, MA, 1983). [7] D.A. Waterman, A Guide to Expert Systems (Addison-Wesley, Reac~g, MA, 1985). [8] M.M. Gupta, A. Kandel, W. Bandler and $.B. Kizka, Approximate Reasoning in Exert Systems (North.HoI~Jd, AmstenJam, 1985). [9] M. Schneider, W. Bandler and A. Kandel, The FESIP Expert System, Proceedings of NAE[PS 85 Worl,.~op on Fuzzy Expert Systems and Decision Support (Oct. 1985).
373
P ~ O P E ~ T ~ E S O F T H E F U Z Z Y EXPEC-WED V A L U E M F U Z Z Y EXPECTED I N T E R V A L ~ Moti SCHNEIDER and Abraham K ~ ' D E L Deparlment of Computer Science and the Institu~ for Expert Systen~ and ~bo~s, Flork~ State Umvers~, Tailahassee, ~ 32306, USA Received April 1986 R e v ~ d July 1986 The evaluation of the fuzzy expected value (~'~V) r©qui'tesa complete knowledge about the domain of the evaluation, and the distribution of the population in that domain. Since it is not always possible to assmne a complete knowledge about the domain, it is necessary to find some relaxations to the ~stdctions involving the evocation of ~ V . In this paper some solutiom ~o tb~ problem are proposed via the concept of the fuzzy expected interval (FEI).
Keywords: Fuzzy expected value, Characteristic functions, Fuzzy expected interval.
1. I a ~ c ~ c f l ~ a
In order to compute the fuzzy expected value ( ~ V ) , as well as any other measure of typicality, it is necessary to know how the population is distributed and what is the grade of membership of each group. In other words the fuzzy expected value may not be applicable when partially unknown or incomplete information is given. In this paper we present several methods to overcome this problem. First we discuss cases in which the fuzzy expected value will not be affected when some incomplete data is utilized. We then proceed to show how to appro~mate the fuzzy expected value in a fuzzy environment by introdudng the fuzzy expected interval. The fuzzy expected interval is found to be useful in many cases where the fuzzy expected value cannot be found. Moreover it wit] be shown that in some cases the fuzzy expected interval can be reduced to a fuzzy expected value even when partially fuzzy information is availab|e~
2. Fu~y Elpeded V~ue Let XA be a B-measurable function such that XA~ [0, I]. The Fuzzy Expected Value (FEV) of XA over the set A, with respect to the fuzzy measure ~(.), is defined as su
lj (rain[r,
* This work is partially supported by NSF grant IST8~5953. 0165-~114/88/$3.50 © 1988, Elsevier S~iencePublishers B.V. (North-Holland)
(1)
374
M. $chnekler, A. Kandel fA(T)
l'k
fA(T)
HI---
0
H
T
1
Fig. 1. The evaluation of FEV(gA).
where
{x [
r}. Now,
T} =fA(r) is a function of the
threshold T [t]. The function ~ maps ~ into the interval [0,1]. The actual calculation of FEV(xA) consists of finding the intersection of the curves T--f.4(T), which will be at a value T = H, so that FEV(gA)ffi H ~ [0, 1] [2]. Figure I depicts this procedure while the following exsmple illustrates how the FEV can be computed. ~p~ L Using the base variable 'hourly wages', assume a given population and a given subjective compatibility curve such that I person earns $3.00--~X = 0 . ~ ,
3 persons 4 persons 2 persons 2 persons
earn earn earn earn
$ 4 . 0 0 ~ X - 0.50, $4.20~X - 0.55, $4.50~ X = 0.60, $10.004 X "- 1.00.
As can be seen we have 5 different thresholds (0.40, 0.50, 0.55, 0.60, 1.00). The first step in the process is to check how many persons are above each threshold (in pex~ntage terms). As can be seen, 12 persons are above or equal to 0.4, 11 persons are above or equal to 0.5, 8 persons are above or equal to 0.55, 4 persons are above or equal to 0.6 ~md 2 persons are above or equal to 1.00. Pairing these data and rearranging by increasing order (of the measure of belief), we obtain the following five [7", ~] pairs (see Eq. (1)): (0.40, 1.00),
(0.50, 0.91),
(0.55, 0.66),
(0.60, 0.33),
(1.00, 0.16).
Now, the minimum value of each pair is: min(0.~, 1.~) = 0.40, min(0.~, 0.33) -- 0.33,
rain(0.50, 0.91) ffi 0.50, rain(1.00, 0.16) - 0.16,
rain(0.55, 0.66) -- 0.55,
.Fuzzy~ z e d v ~ and fuzzyetpec~dintenJal
375
and, therefore, following ~ . (I), the FEV, which is the maximum of all these m i n ~ a , is: max(0.40, 0.50, 0.55, 0.33, 0.16) ffi 0.55.
Thus the FEV is 0.55. From this result we can obtain that the fuzzy expected hourly wage is $4.20. Another approach to find the fuzzy expected value is by arranging all w~lues in increasing order and to find the median. The median is the FEV. Thus
the median is 11/20-0.55 and so is the ~ V . The following theorem proves that the median of the seqeence is the I ~ V .
~ e o ~ m 1. The medi,~n of the set of 2n - 1 numbers, represenffng aU the X~s Cn numbers) and all but t~,efirst ~'s (n - 1 numbers), where 0 <~X~ < X2 < " " < Xn <~ 1 for finite n, arranged'l~n order of magni~de, is the FEV [3]. Establishing the fa~L that the FEV is the median of an ordered list is' very helpful in the developr~lent of new prope~ies about the F]EV. But first we discuss the issue of the characteristic function.
3. The e b M e t e ~ e
~e~m
One of the most important issues involving FEV is choosing the appropriate characteristic function which is used for mapping from the actual values to the interval [0, I]. Choosing the wrong characteristic function may result in wzong interpretation of the mapping, and thus make the whole evaluation of the FEV meaningless. There are two main goals to consider: I. The characteristic function must be monotonic to avoid ambiguity. 2. The range of the characteristic function must be sufficient to cover aU possible cases (and only these cases). ]Example 2 demonstrates how to construct a meaningful characteristic function for a particular population. E _r~m_~e 2. Suppose we have the following characteristic function for the variable OLD: X(x)
~x/lO0 h
if x ~ 100, fix > 100.
From the characteristic function described above it is clear that the function covers all ages and that the mappings are one to one. Thus, it is very important to construct a characteristic function such that the mapping will be one to one, and this function must cover only the relevant cases. By doing so, we can guarantee logical mappin~ from the actual nun~bers to the
M. Schneider, A. Eandd
376
corresponding X's. Next we discuss the concept of the fuzzy expected interval
4. ~ t e r v l
of ~ V
In general, whenever there is a population divided into n groups with n distinct X's we have
different arrangements of the set, and possible 2n - 1 different FEV's. Example 3 illustrates these points. E x ~ p t e 3. Let X be a pop~ation sucE that x people are of age = (compatib|e value X~), y people are of age p (compatible value X2), z people are of age 7 (compatible value X3), where
Z _y+Z
x+y+zfX,
~4
O<~x~
Since n = 3, there are 10 different arrangements as given below. Z2
Zz
z X
z X
y+z X
Xz
X2
3 Xl
~,
Xz
X3
4 Zl
~
X2
g~
,/k
1
Z~
2
z
z
y+z X
Z3 ,y +z
y+z
-~"
X3
Z~
y+z X
Zs
Z~
X2
E~
Z~
X2
z ~
8 x~
X2
~
9 Xl
~
z
y+z
5 6
7
z X Z
z
tO ~
y+
Z~
z
T
y+z
-~-
y+z X
X3
Z
X x3
y+z
X2
X3
x2
X3
It can be seen that xow 1 has FEV = X3, tow 2 has F E V = Zl, iows 3 through 6
F~¢zzyexpec~edvalue and fuzzy expectedintervM
377
have FEV=X2, rows 7 and 8 have FEV = z / X and rows 9 and 10 have F E V -- (y -~ z)]X. Thus the possible number of FEVs is 5, as was stated at the beginning of the section. Now suppose that x people above are of age o? with value X~, such that X~
~f. If both l~ and I* are greater than I. then only the upper n - 1 numbers in L may have different arrangements, and thus I. is still the median. In case both !*~ and 1,,, are less than In then only the lower n - 1 elements in L may have different arrangments, and In will still be the median. Since the median remains the same, so is the FEV. [:] "l%eerem 3. Let L be an ordered set of all X's and ~'s and l~ be the med~n. Then changing 1., to l* ~ {Ira - e, In, + e}, where 1" is some Xj such that l~ - e > Ira-1 and l,,, + e < Ira+l, will not change FEV.
~f. Let Theorem 2 both ~ and change the
a~= l,n - e and ~ = l,n + e. Since ~ is distinct then according to it will not change FEV. We use the same argument for ~5. How, since fl will not change FEV, then any number in the interval [~, IS] will not FEV. El
If the change of 2~i to X* results in X* =X j, then we combine the two populations and reduce n by 1. The importatJce of Theorem 3 is that it allows us to manipulate imcomplete (or fuzzy) information. Example 4 illustrates this point. E ~ m p [ e 4. Suppose we have population A and the following distribution: Pl
Xl
P2
X2
*
m
o
°
Pi
Xi
o
Q
*
o •
Pj
X~
*
•
i
Q
m
m
Pn Xn where A --Pl 4"P2 4--. ° +p~, aIt~ Xi is the FEV. Now substitute pj and £j with pj
378
M, ~hne/der, A./~'~u~el
~ d move or less X~. We ~ h~e~p~et, ~o~¢ [ ~ - e, ~ + e]. Now the new distribution is: P; P2
X~ g2
P~
Z+
•
P,,
~ ~jf
~,
According to Theorem 3 the I ~ V wil not change. Thus it is possible to use some fuzzy informatio~ and still compute the FEV. Moreover, using incomplete hu~ommtio~ will not cha~ge FEV. In conclusion we can state the following corollary: Cen~. ¢ ~ g m g all X's (except the ~ equal to the FEV) to/ntewa/.~ such that ed[ intewols ave disjoint and al~ ~'s ave no~ in d~.se intewals, w ~ not change F~V. ~ e e f o L e t e ] = ~ j - e , Xj+e], j - l , 2 , . . . , 2 n - l , sad let fl]e~], ] I, 2 , . . . , 2n - 1. Then according to Theorem 3, we can change fll to fl~' • ~ and F~V ~ . not be affected. Changing any ~j to fl~ ¢ ~ (such that n :~]) will not change FEV.
The concept of the ~ expected interval was developed in order to cope with incomplete (or fuzzy) information. The ~ I must be interrelated with ~ V in such a way that the result of the evaluation will be either ~ V or FEI but in both e.~es it must have a meaningful mapping back to the o r i ~ a l numbers. In this section we consider the case where the X's may or may not be represented in intervals, however, these intervals must be disjoint. In addition, we allow the/Fs to fall into the intervals of the X°s. Consider the following example. Emmple $, Let A be a population such that x people are within the age range of ~ to ~2 (computable values Xt to X2), y people are within the age. tango of ~ to f12 (computable values X3 to X4), where A = x + y and X~< X~< X3 < X4. Now we have the foHowhlg arrangements:
I
x2], L 3- x,,]
Fuzzy expec~d v~l,ueaml fuzzy expeaed t~ceruM
3
l'l
4 Lxt -" X2], L~3- g4] when y falls into one of the intervals. First 3 cases present no problems, s ~ we can use Theorem 1 and make the median the FEV (case 2) or FEI (eases 1, 3). But what about case 4? Since in e.~l~4 th¢l¢ is no median we have to seek a differem solution. O~xecan see that if y/A falls inside the int©rval L~, X2] i~ is reasonable to assume that the FEX will be in that in,real. The same argument may be applied to the case where ylA fails into the interval IX3, X4]. To make the decision process systematic we can define the following role (based on Theorems 2, 3 and the corollm'y): R ~ 1. Whenever some ~ falls inside a given interval, delete the $, and duplicate th© interval. In conclusion, we can say that using incomplete information doesn't have to d ~ ' t our decisions. By providing a complete characteristic function and some information about the population (or the domain in which we have to make a decision), the F~V or the ~lEI can still be computed. Consider the next example. 6. Let the characteristicfunction of the variable O L D be:
x(x) =
/1oo
ifx <0, if x ~<100, if x > 100,
and rite dis~bution of the population is 10 people are of age range 10--20, 25 people are of age 30, 15 people are of age 40, 30 people are of age range 45-55, 20 people are of age range 60-70. Now, translating the actual numbers to ~'s and g's we get
1,00 0.90 0.65 0,50 0.20
0.1-0.2 0.30 0.40 0.45-0.55 0.6--0.7
M. Schneider, A, Kandel
380
Ushg ~
'"
[oa-0.2],0.3,
[0.6-o.7],[0.6-0.7],0.90
and the median is [0.45-0.55].Thus the FEI is [0.45-0.55].Going back to the originalnumbers we conclude that the fuzzyexpected age ~ between 45 and 55. It is easy to see that if we had 0.55 instead of the range [0.45-0.55] then the median would be 0,5 and the fuzzy expected age would be 50. Now we can present the theorem about the relation between FEV and FEI.
~ e o r e m 4. Let L ffi {11,. • •, l , , . . . , 12,-~} be an ordered list of I~'s and X's such that any X~ can be represented by either a single valued number or by an interval. Then FEV ¢ FEI. Proof. Let FEI = { s ~ , . . . , s~} be ~he median of L. Then FEI must contain the FEV, since any selected s~ ~ FEI will remain the median, and thus s~ -- l,, -FEV ¢ FEI. D Next we consider the case where the intervals may overlap. 6. ~
the FEV ar ~he FEI in o v e ~ p ~
~tervgq
The problem with intervals which are overlapping is how to deal with the case where a certain/~ may fall within the intersection of two or more intervals. In this case Rule 1 will not be applicable since Rule 1 is appropriate only for cases in which the intervals are mutually exclusive. One possible solution to this problem is to break the intervals in such a way that the intervals will become disjoint. Example 7 illustrates this point. ~~e
7. Let A be a population and let the distribution of A be: X people are within the age range of ~r1-¢2, Y people are within the age range of ~ 3 - ~ ,
such that ~1 < ~3 < c¢2< ~4 and A = X + Y. The regular solution will be to find all possible arrangements. Now we can break the population in the following way: xl people are within the age range of ah and c¢3 - e, X - x t + Y - y a people are within the age range of at3 and o:2 - e, Yl people are within the age range of c~2 and ~4. such that X = x~ + x2, Y = yl + y~. and ~:he intervals are disjoint. Assume that a relevant characteristic function generates the following mapping: ~:"> Xl,
~3 - e--->X2,
~:"> X3,
Fuzzy expected value and fuzzy expec~d ~ r v a l
381
Now we ¢a~ pair the data:
X - xt + Y - y~ + yl A Yl
L% - X~]
L~-~,]
We can see the problem. Since we have no information about x~ and y~ we can not find the FEV nor the FFI.
It is clear that we have to find a differem Way to get around this problem. Thus we have to establish a rule to handle cases where a given ~ may fall in more than one interval. (The following rule is established emph~cally, the justification will follow in Theorems 5 through I0.) Rule 2. Whenever a certain/~ falls in more than one interval, choose the interval such that the ~ falls as close to its nfiddle as possible. After choosing the hlterva~, delete the ~ and duplicate the interval T ~ o u g h the use of Rule 2 we can now solve the problem in Example 7 by creating the following arrangements (assuming the mapping: ~--> X~):
-Y L~-- x~], L~3 - x,], A' Y
[X~ - X2], [Z~ - X2], [Za - X4] if Y is closer to the rlliddle of the f ~ t interval, .Y [Z1 - X2], L~3 - x4], L~a - x4] if ~ is closer to the middle of the second interval. We can formalize the rules by writing the next six theorems.
Def~liom 1. Let S and R be two intervals such that $ = {sl, .... ,s.} and R = { r ~ , . . . , r,,,}, then: 1. MAX(S,R) ffi $ ff for every s~ ¢ S there is rj ¢ R such that s~ > rj. 2. t~rN($, R) = S ff for every s~ • S there is r] ~ R such that ss < rj. Note. We assume that each interval may contain a single element,
" £ h ~ m S° Let $ and R be two iaterva~ such taut (a) S = { s ~ , . . . , s,}, (b) R = { r ~ , . . . , rm}, and (c) R n S = ~. Then
~(S,R)=
{~ ifr,>s.. if s~ > r,,,.
(3)
382
~f~
M. Schnader, A, Kandd
Let s~ ~. r~.. Then . . b~sed ~n D e f i ~ f l ~ lqL~, ~.,..,_x($,R) - S,. []
~e~m R ffi { r ~ , .
6. Let $ and ~ be avo intervals such that (a) $ ffi { s ~ , . . . , s,}, (b) . . , r m } , and (c) R N $ = ~ . Then {~ ~ ( $ , R) ffi
ifrm
(4)
~ o e f . Let r,,, < s~. Then based on Definition 1(2), urn(S, R) ffi R. Theorem 7. Let S and R be two intervals such that (a) S ffi { s ~ , . . . , s , } , (b)
Rf{rt,...,r,,,},
(c) R N S ~ ,
~L.x(& R) ffi {R
(d) $ ¢ R a n d R ~ $ ,
a n d ( e ) s n > r ~ . Then
ifrm>s,, if s~ > rm.
(5)
Proof. Let rm > s,,. Then based on Definition 1(1), ~o,x(S, R) = R.
r3
"~eorem 8. Lea S and R be two imervds such that (a) S ffi { s l , . . . ~ an}, (b)
R f f i { r l , . . . , r ~ } , (c) R N S ~ , ~ ( $ , R ) _ _ {R
(d)$¢RandR¢$,
o~d (e) s, >r~. Then
ifr~
(6)
if s. < r..
Proof. Let r~ < s,. Then based on Definition 1(2), ~L~($, R) ffi R.
D
To make the theory complete we need the foUowing definition. De~,1eon 2. Let $ end R be two intervals such that $ = { s l , . . . , s,} and R ffi ( r ~ , . . . , rm} and R _=$. Then there is an interval T such that: 1. uAx($, R, T) = 7", ff for every t~ ¢ T t~ere is sj e S such that tj ~ sj and for every ta ¢ T there is vk E R such that ta ~ r~. 2. ~ ( $ , R, T) = T, if for every t~ ¢ T the,re is sj e $ such that t~<~sj and for every tte T there is rk ¢ R such that tt ~ rk. T h e e r e l 9. Let $ and R be two intervals' such that 3 ffi {sl, . . . , s,} and
R ffi {v~, . . . , vm} and R =_$. Then
MAX(S,R) {rl,..., s.). =
(7)
P ~ f . Let T be an ordered set such that T - { r x , . . . , s,}. Then based on Definit/on 2(1), MAx(S, R, T) -- 7". 0
Theorem 10, Let S and R be two intervals such that S ffi{s~,..., sn} and R - { r ~ , . . . , rm} and R =_S. Then MIN(S, R ) =
. . . , rm}.
(8)
Fuzzy expecteduMueand fuzzy expectedintevual D e f ~ o n 2(2), Mr~q($,R, T) = Z
383
El
Defe~lieu 3. Let ¢ and tff be intervals. Then we say that ¢ i~ higher than ~ ff the upper bound of ¢ is greater than the upper bound of ~. KvJ~ple 8. Using F,q. ( l ) and the data from Example 6, we can form the following pairs: /A
Z
1.00 0.75 0.65 0.50 0.20
0.1-0.2 0.30 OAO 0.45-0.55 0.6-0.7
Now, according to Eq. 1 we have to find the M=N for each pair:
~N(l.oo, [o.l-o.21) = [o.l-o.2]
(Eqo 4),
MIN(0.?5, 0.3) = 0.3, ~uN(0.65, 0.4) = 0.4,
~(0.5, [0.45-0.55]) = [0,45-0.50] ~uN(0.2, [0.6-0.7]) --- 0.2 (Eq. 4).
(F I. 8),
The next step is to find the MA× over all the ~Ns:
~x([0.1-0.2], 0.3, 0.4, [0.45=0.50], 0.2)-- [0.45-0.50]. The following algorithm summarizes the steps involved in the computation of the FEV or the FEI: 1. Find eli the ~'s and X's using Eq. (1) and the characteristic function, respectively. 2. Find the win of each pair using Eqs. (4), (6) or (8). 3. Order the results above into an ordered list L based on Definition 3. 4. Find the wax of L utilizing Eqs. (3), (5) or (7). The following example demonstrates that the tools introduced in this paper allow the possibility of computing FEV in spite of the fact that all Z's are given as intervals. Ex!mp|e 9. Let the characteristic function of the varieble OLD be:
x(x) =
/loo
if x < O, if x ~<100, if x > 100,
384
~
M. Sc3me~,v, A./Candd
~r~ ~ b ' ~ o n
ef the population be:
30 people 20 people 30 people 20 people
are are are are
of the of the of the of the
age age age age
ranging from ranging from ranging from ranging from
10 to 40 to 50 to 80 to
20, 50, 70, 90.
According to step 1 of the algorithm described above, the following pairs are computed: 1.00 0.70 0.50 0.20
0.1-0.2 0.4--0.5 0.5-0.7 0.8-0.9
Now, the ~NS Of the pairs are (based on step 2 in the algorithm): ~s(1.00, [0.1-0.2]) = [0.1-0.2] ~uN(0.70, [0.4--0.5]) = [0.4-0.5] u (o.5o,
- o.5
(Eq. 4), (Eq. 4), g),
~ ( 0 . 2 0 , [0.8-0.9]) = 0.2 (Eq. 4). Thus the ordered Hst of the Mn~scomputed above is:
L-- {[0.1-0.2], 0.2, [0.4-0.5], 0.5} Now, according to the algorithm the maximum of L is MAX([0.I-0.2], 0.2, [0.4--0.5], 0.5) = 0.5 (Eq. 7). ?. C e a e ~ l e n In this paper we have ir:~oduced some issues related to the concept of the fcz~ expected value and the theory of typicality, it is shown that by adding the concept of the fuzzy expected interval we can solve some interesting problems involved in these issues. The use of the fuzzy expected interval is found to be very useful when the information regarding the domain and the distribution of the popu|ation is partially unknown or incomplete. Moreover, in some cases we are able to reduce the s~e of the interval and map that interval into a single valued number when going back from the characteristic function to the original data. The fact that we can compile incomplete data is quite usefu! in dealing with fuzzy variables and building fuzzy expert systems [4--9].
Rdem [1] A. ~nde! and W.L Byatt, Fuzzy sets, ~uzzy algebra, and fuzzy stads~cs, Prcc. IEEE 66 (i2)
Fuzzy expected value and fuzzy expectedinterval
385
I21 A. KandeL Fuzzy Te~eniques in Pattern Recognition (Wfley-Interscieuce, New, York, 1~2). [3] A. ~tndei, Fuzzy Mathemaucal Techniques v~th App|ications (Addi~n Wesley, Rea~ng, MA, 1986). [4] C.V. Negoita, Expert System and Fuzzy Systems (Benjarain/Cummln~, New York, 1985). [5] S.M. Weiss and C.A. Kulikowsld, Designing Expert Systems (Rowman and A~anheld, Totowa, NJ, 1984). [6] F. Hayes-Roth, D.A. Waterman and D.B. Lenat, Building Expert Systems (Addison-Wesley,
Reading, MA, 1983). [7] D.A. Waterman, A Guide to Expert Systems (Addison-Wesley, Reac~g, MA, 1985). [8] M.M. Gupta, A. Kandel, W. Bandler and $.B. Kizka, Approximate Reasoning in Exert Systems (North.HoI~Jd, AmstenJam, 1985). [9] M. Schneider, W. Bandler and A. Kandel, The FESIP Expert System, Proceedings of NAE[PS 85 Worl,.~op on Fuzzy Expert Systems and Decision Support (Oct. 1985).