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Entropy variation on the fuzzy numbers with arithmetic operations
sets and systems ELSEVIER
Fuzzy Sets and Systems 103 (1999) 443-455
Entropy variation on the fuzzy numbers with arithmetic operations 1 Wen-June Wang*, Chih-Hui Chiu Department of Electrical Engineering, National Central University, Chung-Li 320, Taiwan, ROC
Received October 1996; received in revised form April 1997
Abstract Entropy is always the measure of the fuzziness degree for a fuzzy set. This paper studies the entropy variation on the fuzzy numbers (or fuzzy sets) under the following several cases: (i) two fuzzy sets on the same type but different support size; (ii) the addition of finite number of fuzzy numbers on the same type; (iii) the subtraction of finite number of fuzzy numbers on the same type; (iv) the multiplication of a constant and a fuzzy number. It is shown that through the above arithmetic operations, the resultant fuzzy number is the same type as the original fuzzy numbers and the entropy of the resultant fuzzy number has the arithmetic relation with the entropy of each original fuzzy number. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Entropy; Fuzzy numbers; Measure of fuzziness; Arithmetic operations
1. Introduction In general, the membership function of a fuzzy set is determined by users subjectively. The shape of a membership functions always presents the knowledge grade of the elements in the fuzzy set. In other words, every membership function also presents the fuzziness of the corresponding fuzzy set in the idea of users. Therefore, it is necessary having some measurements to measure the fuzziness of a fuzzy set. Until now, there have been several typical methods being used to measure the fuzziness of fuzzy sets. De Luca and Termini [2] utilized the conception of the "entropy" to indicate the fuzziness of a fuzzy set. K a u f m a n n [7] proposed that the fuzziness of a fuzzy set can be measured through the distance between the fuzzy set and its nearest non-fuzzy set. Yager [13] suggested the measure of fuzziness can be expressed by the distances between the fuzzy set and its complement. We can not deny that the entropy is indeed a proper measurement of a fuzzy set and has received a lot of attention recently. There is lots of literature talking about the entropies of fuzzy sets (see [1, 3, 4, 9, 12]).
* Corresponding author. E-mail: [email protected]. 1This work was supported in part by the National Science Council of Taiwan, R.O.C. under the Grant NSC 85-2213-E008-019. 0165-0114/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S01 65-0 1 14(97)001 82-6
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Pedrycz [lo] is the original motivation of this paper. Pedrycz [lo] showed the entropy change when the interval size of the support is changed. He considered the fuzzy set with triangular membership function. In this paper, the result of [lo] is extended to the other types of fuzzy sets. Next, we consider three common types of fuzzy numbers through arithmetic operations. These operations include addition, subtraction and multiplication. The resultant fuzzy number after arithmetic operations has its entropy. What is the relationship between the entropies of the resultant fuzzy number and the original fuzzy numbers is the main task of this paper. For the triangular fuzzy numbers, these results has been proposed in [l 11. However, this paper extends the consideration to the other three common types of fuzzy numbers. This paper is organized into five sections. In Section 2, we introduce some properties of the entropy. In Section 3, we discuss the relationship of the entropies between the fuzzy number after arithmetic operations and the original fuzzy numbers. Some examples are illustrated in Section 4. Finally, a brief conclusion is included in Section 5.
2. Preliminary A fuzzy set A is defined in a universal set X, where X is real and finite. Let A(x): x + [0, l] be the membership function of the fuzzy set A for x E X. Let a measure of fuzziness be denoted by H(A) and have the following properties [2]: 1. H(A) = 0, if A is a crisp set in X. 2. H(A) is a unique maximum if A(x) = *, Vx E X. 3. For two fuzzy sets A1 and AZ, if A,(x) < A,(x) for A,(x) < f and A,(x) > A,(x) for Al(x) 2 i then H(A1) 2 H(&). 4. H(A”) = H(A), where A” is the standard complement of A, i.e., A”(x) = 1 - A(x). As mentioned in [2] the measure of fuzziness H(A) can be regarded as an “entropy” of the fuzzy set A. At a fixed element x,
WA(x))= W(x)),
(2.1)
where the entropy function h: [O, l] + [0, l] is monotonically increasing in [O,i] and monotonically decreasing in (i, 11; moreover, h(u) = 0, as u = 0 and 1; and h(u) = 1, as u = 3. Some well-known entropy functions are shown in the following: if u E [0, 31, 2u 2(1 - u) if UE [$, 11,
I
(2.2)
h(u) p 4u(l - u).
(2.3)
h(u) p - u lnu - (1 - u)ln(l - u),
(2.4)
h(u) p
and
where (2.4) is called Shannon’s function [15]. To determine a global entropy H(A) of the fuzzy set A independent of x, we can integrate over the universal set X as follows [lo]: H(A) 4 s XSX
h(A(x))&)dx,
(2.5)
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445
where p(x) is the probability density function of the available data in X. It is known that the larger H(A) is, the more is the fuzziness of the fuzzy set A. Now, we will show that the global entropy H(A) depends on the size of the support of the fuzzy set A. For the membership function of the fuzzy set A(x), A =denotes the or-cut of A, i.e. A ~ ~- {xlA(x) >>.~, x eX}; A"" is the strong or-cut of A, i.e., A ~+ ~- {xlA(x) > ~, x e S }. If e = 0, A °+ is called the "support" of A. Definition 1. Suppose we have any two fuzzy sets A1 and A2 with the support A °+ = (al, bl) A°+ = (a2, b2) = X, respectively. If A1 (~) = A2(~),
c
X and
(2.6)
where = a l + c1,~ =a2 +(Ca/(bl - al)) (b2 - a2), 2 e [ax, bl] and )~e [a2, b2]; or
Ax(Y) = A2(~),
(2.7)
where x = a l - 4 - C l , £ = b 2 - ( C l / ( b l - a l ) ) ( b 2 - a 2 ) , x 6 [ a l , b l ] and £ e [ a 2 , b2]. Then we call A1 and A2 to be the "same type of fuzzy sets". (a) and (b) of Fig. 1 illustrate the forms (2.6) and (2.7), respectively. It is also noted that if b2 = - al, a2 = - bl then let A2 be called the "image" of A1 and be denoted by Ai(see Fig. 2), where A1 and A2 also satisfy (2.7). o+ where the "size" of A °+ is defined as It is obvious that the size of (A~-)°+ is equal to the size of A1, bl - a l .
~'~
bl
f
0
aa
b2
22,
(a)
a~
~,~
b~
a2
Co)
Fig. 1.
b2
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W-J. Wang, C.-H. Chiu / Fuzzy Sets and Systems 103 (1999) 443-455
al
b~
~-
~
0t
Fig. 2.
L e m m a 1 (Chen and W a n g [1]). For the above same type of f u z z y sets A 1 and A 2 satisfying (2.6), suppose p(x) = s, where s is a constant over [al, b l ] u [ a 2 , b2], then H(A~) = kH(a2),
(2.8)
where k = (bl - al)/(b2 - a2) and H(Ai) z~ Sb, p(x)h(ai(x))dx, i = 1, 2. R e m a r k 1. L e m m a 1 states that ifAx is "fatter" (or "thinner") than e n t r o p y of A~ is k times of the e n t r o p y of A2.
A 2
with the ratio k > 1 (or k < 1) then the
R e m a r k 2. This l e m m a does not constrain that Ai, i = 1, 2, must be convex, piecewise linear, n o r m a l or a specific type. F r o m L e m m a 1, we can get the following result easily.
Theorem 1. For a f u z z y set A and its image A - , we have (2.9)
H(A) = H ( A - ) .
Proof. F o r the two fuzzy sets A and A-, where the support A °÷ = (a, b) c X and ( A - ) °+ = ( - b, - a) c X, we have A(2) = A - (~), where ~ = a + k, ~ = - a - k. Thus, we can use the same way of the p r o o f of L e m m a 1 to prove the result (2.9). [] Corollary 1. Suppose two f u z z y sets Ax and A 2 satisfy (2.7); then (2.10)
H(A1) = kH(A2), where k and H(Ai), i = 1, 2, are defined in Lemma 1.
Proof. Since T h e o r e m 1 says that H ( A 1 ) = H(A?). By L e m m a 1 and (2.7), we have H ( A ? ) = kH(A2). Therefore, it is obvious that (2.11) holds. [] R e m a r k 3. F o r the aforementioned two fuzzy sets A 1 and A 2 satisfying (2.6) or (2.7), if b 2 it does not need that al = a2 and bl = b2, the global e n t r o p y H(A1) is equal to H(A2).
a 2 =- b l -
al,
but
W.-J. Wang, C.-H. Chiu / Fuzzy Sets and Systems 103 (1999) 443-455
447
3. The entropy change through arithmetic operations Let us recall the notations of arithmetic operations on fuzzy numbers. A fuzzy set A is called a "fuzzy number" if it is convex and normal [6]. Suppose A, B are two fuzzy numbers; we have [8]
(A®B)~ z~ A ~ ® B ~, as(0, 1];
(3.1)
moreover,
xCA ~x.~A~'
~A(x)~{O
Then by the decomposition theorem [8], we have
A®B~- U ~(A®B),
(3.2)
~C0, 1]
where ® denotes any arithmetic operation which may be addition, subtraction, multiplication, and division. 0 denotes the standard fuzzy union. We consider here three basic arithmetic operations: addition (i.e. A + B), subtraction (i.e. A - B), and a simple multiplication (i.e. k" A where k is a constant). It is known that if the fuzzy numbers A and B are triangular forms then A + B, A - B, and k. A, k ~ R, are also triangular fuzzy numbers [6]. For a set of triangular fuzzy numbers Ai, i = 1, 2, ..., n, and Bj,j = 1, 2 . . . . . m, the entropies of H(~iAi®ZjBj), Y.iH(Ai), and y, jH(B~) have been studied in [11]. Here, we try to study the entropy variation for the other types of fuzzy numbers with arithmetic operations. Let us consider several fuzzy numbers with different type of membership function as follows [8]: 1
Type (i): Al(X) =
+ pl(x - r) 2'
x~A~, (3.3a)
elsewhere,
Type (ii): A2(x) =
{~
-Ip2(x-r)r xeA~2, elsewhere,
(3.3b)
where e is a very small fixed constant, i.e. 1 >> e > 0. Moreover, A~=
r-
+ ~/
Pl
Type (iii): Aa(x)=
~
- Pl 2
in(3.3a) and
A~=[r+(lne/p2),r-(lne/p2)]
in(3.3b).
_1
i 1 11
x ~ r - --., r + P3 elsewhere,
, (3.3c)
where r e R for which the membership grade is required to be one; Pi > 0, for i = 1, 2, 3, is a parameter to determine the rate at which, for each x, the function decreases with the increasing difference Ir - x l. Remark 4. In order to give a finite size of the supports (bounded support) of Type (i) and Type (ii) fuzzy numbers, we have to define Ai(x) = 0 as x¢A~ i = 1, 2, in (3.3a) and (3.3b).
W.-J. Wung, C.-H. Chiu / Fuzzy Sets and Systems 103 (1999) 443-455
(a) A,(x) in (3.3a) with ~2 and e = 0.027 .(b) A,(x)in (3.3b) with ~2 and E = 0.05.
(c) As(x
(3.3~) with r=2. Fig. 3.
Next, we. will show that A and B are the fuzzy numbers with the same type of membership function then A + B, A - B, and k. A, kE R, are also the same type as A and B. For the fuzzy numbers of Type (i), A,(x) = l/(1 + pI(x - r#) and AI(X) = l/(1 + p~(x - rz)‘); then by (3.1) and (3.2) we have
1 (A, + AI)(x) = i l0
1 + -$(x - (rl +
XE (A, + Ail)&,
elsewhere;
1 (A, - A&(x) =
(3.4a)
r2V
1 + -$(x - (rI 0
XE(A, - A$, (3.4b)
r2))’
elsewhere;
W.-J. Wang, C-H. Chiu / Fuzzy Sets and @stems 103 (1999) 443-455
449
and x e (kA1) ~, el
(3.4c)
1 + - ~ (x - - krl) 2
kAl(x) =
elsewhere;
0
where q --&~
+ v/Up2. Moreover, by a simple calculation, in (3.4a)
(A1+,41)'=
(rl+r2)-
1 +
e,(rl+r2)+
+ / x/
=
--
--
d- ~/P2)
N/
/3
(rl
-- r2) +
-I-~p2f~l~
e
I
1
in (3.4b); and in (3.4c)
It is seen that A1 + ,41, A1 - - 4 1 and k ' A l are the same type as A1 and .4~. The detail derivation for (3.4a)-(3.4c) is shown in the appendix. For the fuzzy numbers of Type (ii) or Type (iii), we also can have the similar derivation for them to get the same type A~ + Ai and k.A~, i = 2, 3. For a set of fuzzy numbers Ai, i = 1, 2 . . . . . n, and Bj, j = 1, 2, ..., m, next will discuss the entropies H ( ~ A ~ ® ~.jBj), ~ H ( A ~ ) , and 2 j H ( B j ) . 3.1. Addition
First, let ® be the addition " + " and let the support A°+= (a,, b~) c X. Suppose
C = ~, A~.
(3.5)
i=l
Note: For the fuzzy numbers of Type (i) and Type (ii), A °+ = A~+, i --- 1, 2.
Here, the membership function of Ai, i = 1, 2, ..., n belongs to the one of the three Types (i), (ii), and (iii), n n C o+ = (c, d) c X, where c = Yi= 1 a~ and d = y~= 1 bi. From (3.4a) and appendix, it is easy to see C and A~, Vi, are same type then C(£) : A,(~), i = 1, 2, ..., n, where k
= ai + k, £ = c + 7 - - - -
oi -- ai
(d - c).
Therefore, the following result is given.
(3.6)
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450
Theorem 2. For the above f u z z y numbers C and As in (3.5), suppose p(x) = s, where s is a constant over X , we have
(3.7)
H(C) = ~" H(A,). i=1
where H(" ) is defined in (2.5).
Proof. From (3.6), we have Ai(x)=C
)
c +~(x-ai)
=C(y),xe[ai,
bi],i= l,2, ...,n,
where y = c +
d-c
bi - ai
(x - ai).
Since d y / d x = (d - c)/(bi - ai), then tt(A1) + H(A9
-
=
bl - al c
+ ... + I-I(A.) =
s'h(C(y))dy-+
fabl s . h ( A l ( x ) ) d x 1 b2 - a2 c
+
fab2s . h ( A 2 ( x ) ) d x 2
s.h(C(y))dy+
+ ... +
f:" s . h ( A n ( x ) ) d x n
bn - an ;a ... "4-----Z--~cd~d s'h(C(y))dy
(bl + b2 + ... + bn) - (al + a2 + ... + an) ( e Jc s" h(C(y)) dy d-c
= fcd s" h(C(y)) dy = H(C).
From (3.5), this implies that
i=1
i=1
3.2. Subtraction
Next, let ® denote the subtraction" - ". Suppose we have two fuzzy numbers B~ and B2 on the same type (either Type (i), Type (ii) or Type (iii)). From the decomposition theorem, BI - B2 = B1 + BE still holds for the above three types of fuzzy numbers. Therefore, the following result is obvious. Theorem 3. For the above two f u z z y numbers B1 and B2, suppose p(x) = s = constant over X; we have H ( B , - B2) = H ( B , ) + H(BE).
Proof. Since B, - B2 = B~ + B~. By Theorems 2 and 1 H(B,
-
BE) = H ( B , + B~) = H ( B O + H ( B ~ ) = H(B1) + H(B2).
[]
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451
Corollary 3. For aforementioned fuzzy numbers Ai, i = 1, 2, ..., n, in (3.4), we have H(A~ -- A2 . . . . .
(3.8)
A~) = ~ H(Ak), k=l
H(i~
Ai - j ~
(3.9)
AJ) = k=l~ H(Ak)
where I1 and I2 are two crisp sets of digits and I 1 u I 2
=
{1, 2, ... ,n}, Ilf312 = O.
Proof. By Theorems 2 and 3, it is easy to prove (3.8) and (3.9).
[]
3.3. Multiplication of a fuzzy number by a constant Now, we like to discuss the entropy ofk" A, where k is a constant and A is a fuzzy number belonging to any one of the above three types. Here, the scale k can be considered as a fuzzy number K that has the membership value K(x) = 1 as x = k and the membership value is zero for x ~ k. Thus, k" A can be seen as a multiplication between two fuzzy numbers K and A.
Theorem 4. For a constant k and a fuzzy number A as above, we have H(k.A) = k.H(A).
Proof. Suppose the support A °+ = (a, b) c X and C = k" A then it follows C O+= (c, d) c X, where c = ka and d = kb. Moreover C(kx) = A(x). Since c - d = k(b - a), by Lemma 1, it is easy to get H(C) = k'H(A). That means we have H(k.A) = k.H(A). [] Remark 5. F r o m the above analysis, it is seen that the fuzziness increases if several fuzzy numbers go through the addition and subtraction operations.
Remark 6. It is known that the considered fuzzy numbers here are all symmetric (see Fig. 3). If we consider triangular fuzzy numbers which may be not symmetric. Their entropy relation between the resultant fuzzy number and original fuzzy numbers through arithmetic operations needs further discussion [11]. Ref. [11] shows that, in fact, Theorems 2, 3 and 4 are also applicable to triangular fuzzy numbers.
4. Three examples Example 1. Consider two fuzzy numbers A and B of Type (i) as follows:
A(x) =
B(x) =
{1
+ (X -- 3) 2
{ 1
x ~ A °'°1 = [3 -- V/-~, 3 + V / ~ ] , elsewhere;
+ 4(x - 5) 2 elsewhere.
452
W.J.
Wang, C.-H. Chiu /Fuzzy
Sets and System
103 (1999) 443-455
By arithmetic operation, we have 1
x+t+B)O.O’=
1 +;(x - 8)’
(A + B)(x) =
elsewhere; 1
x E (A + B)O.O’=
-2+@9,
1 + ;(x + 2)2
(A - B)(x) =
0
1
elsewhere; 1
(2/I)(x) =
-2-;JG,
x E(~A)O.‘~= [6 - 2,/%, 6 + 2 @I,
1 +;(x - 6)2 0
elsewhere.
Let us take the entropy function as (2.3) and the entropy as (2.5) with p(x) = s, where s is a constant over X. Then, we get H(A) x 5.484s, H(B) z 2.742s, H(A + B) x 8.226s, H(A - B) x 8.226s, H(2A) z 10.968s. According to above result, we know that H(A + B) = H(A - B) = H(A) + H(B) and H(2A) = 2H(A). Example 2. Consider two fuzzy numbers A and B of Type (ii) as follows: exp-lx-31
A(x) =
o
B(x) =
;
-M*
xeAO.O’ = [3 + ln(O.Ol), 3 - ln(O.Ol)], elsewhere;
- 5)l
x E
B”.” = [5 + iln(O.Ol), 5 - ~ln(O.Ol)],
elsewhere.
Then, we have A + B, A - B, and 2A as follows: exp-IWK-88)l
(A + B)(x) =
i
o
XE(~ + qO.01 = [8 + $ln(O.Ol), 8 - $ln(O.Ol)],
elsewhere; = [ - 2 + jln(O.Ol), - 2 - +ln(O.Ol)],
(A _ B)(x) =
(2AJ(xl =
~-“1’2”‘-6”
;l;JQc,“’
= [6 + 2 ln(O.Ol), 6 - 2 ln(O.Ol)],
Suppose the entropy function (2.3) is chosen and by (2.5) with p(x) = s, where s is a constant over X. We can get H(A) x 3.92s, H(B) x 1.86s, H(A + B) w 5.48s, H(A - B) x 5.48s and H(2A) x 7.84s. So, it is obviously H(A + B) = H(A - B) = H(A) + H(B) and H(2A) = 2H(A).
W.-J. Wang, C.-H. Chiu / Fuzzy Sets and Systems 103 (1999) 443-455
Example 3.
453
We have two fuzzy numbers A and B as Type (iii) follows:
[01+ cos((x - 3)~)]
A(x) =
2
x e [-2, 4], elsewhere;
B(x) =
i1 + cos(2(x - 5)n)] 2 elsewhere.
A + B, A - B, and 2. A are given as follows:
(A + B)(x) =
[1 + cos(~(x - 8)n)] 2 elsewhere,
0
[1 + cos(~(x + 2)7t)] (A - B)(x) =
2
-
½],
elsewhere;
0
2-A(x) =
xeE -},
_[01+ cos(½(x - 6)n)] 2
x e [4, 8], elsewhere.
Let us choose the entropy function as (2.3) and the entropy as (2.5) with p(x) = s, where s denote a constant over X. We get H ( A ) = s, H ( B ) = ½s, H(A + B ) = H ( A - B ) = ~s and H ( 2 A ) = 2s. It is obvious that H(A + B) = H(A - B) = H(A) + H(B) and H(2A) = 2H(A).
5. Conclusion This paper has presented several results about the entropy variation of fuzzy numbers. First, the entropies of two same type of fuzzy sets have a constant gain relationship. The gain depends on the sizes of their supports. Second, a fuzzy set and its image have the same entropy. Furthermore, for any one type (i), (ii) or (iii) fuzzy numbers, after some arithmetic operations, their type is kept and the entropy of the resultant fuzzy number can be obtained by calculating the individual entropy of the original fuzzy numbers.
Appendix Here, we prove that if the fuzzy numbers A~ and ,4~, i = 1, 2, 3, are one of the above three types then At + Ai, A~ - Ai and k . & , k e R , are also same type as Ax and A1. First, we consider the fuzzy numbers of Type (i). According to arithmatic operations of intervals, we have
[X//1--~
1~-]1--~ ~.~=E~//1-~
~-I1--~
454
W.-J. Wang, C.-H. Chiu / Fuzzy Sets and Systems 103 (1999) 443-455
So, we can get (A1 +'41)~ --- [ ( r l + r2)
--
(X/~I
q-
~2)~l-ct
, (ri
-4- r 2 ) -b
(A1 A1)a I(rl r2)( 1 (kA1)~= krl
-k
1-
Pl~
,kr~ +
( X / ~ 1 q- . / --i- - ~ . 1-~ -] ~/P2,/~/ ~
;
_[
1 - -
~] Pl ¢z A"
By decomposition theorem, we obtain
{
(A1 + A1)(x) =
1
xe(A1 + At) ~,
+ 1 (x - (rl + r2))2 elsewhere;
xe(A1 (A 1 _ 21)(x) --
-
~Zli)e ,
-4- 1 (X -- (rl -- r2)) 2
elsewhere; and
kA1 (x) =
10+ Pl-~ (x 1
krl) 2
x e (kA 1)~,
elsewhere; where q ~- ~/1/pl + 1V/~2. It is easy to see that Al + -41, Ai - .4~, and k ' A 1 are the same type as A1 and A1. Easily, Type (it) and Type (iii) have the same conclusion by the similar derivation.
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Fuzzy Sets and Systems 103 (1999) 443-455
Entropy variation on the fuzzy numbers with arithmetic operations 1 Wen-June Wang*, Chih-Hui Chiu Department of Electrical Engineering, National Central University, Chung-Li 320, Taiwan, ROC
Received October 1996; received in revised form April 1997
Abstract Entropy is always the measure of the fuzziness degree for a fuzzy set. This paper studies the entropy variation on the fuzzy numbers (or fuzzy sets) under the following several cases: (i) two fuzzy sets on the same type but different support size; (ii) the addition of finite number of fuzzy numbers on the same type; (iii) the subtraction of finite number of fuzzy numbers on the same type; (iv) the multiplication of a constant and a fuzzy number. It is shown that through the above arithmetic operations, the resultant fuzzy number is the same type as the original fuzzy numbers and the entropy of the resultant fuzzy number has the arithmetic relation with the entropy of each original fuzzy number. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Entropy; Fuzzy numbers; Measure of fuzziness; Arithmetic operations
1. Introduction In general, the membership function of a fuzzy set is determined by users subjectively. The shape of a membership functions always presents the knowledge grade of the elements in the fuzzy set. In other words, every membership function also presents the fuzziness of the corresponding fuzzy set in the idea of users. Therefore, it is necessary having some measurements to measure the fuzziness of a fuzzy set. Until now, there have been several typical methods being used to measure the fuzziness of fuzzy sets. De Luca and Termini [2] utilized the conception of the "entropy" to indicate the fuzziness of a fuzzy set. K a u f m a n n [7] proposed that the fuzziness of a fuzzy set can be measured through the distance between the fuzzy set and its nearest non-fuzzy set. Yager [13] suggested the measure of fuzziness can be expressed by the distances between the fuzzy set and its complement. We can not deny that the entropy is indeed a proper measurement of a fuzzy set and has received a lot of attention recently. There is lots of literature talking about the entropies of fuzzy sets (see [1, 3, 4, 9, 12]).
* Corresponding author. E-mail: [email protected]. 1This work was supported in part by the National Science Council of Taiwan, R.O.C. under the Grant NSC 85-2213-E008-019. 0165-0114/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S01 65-0 1 14(97)001 82-6
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Pedrycz [lo] is the original motivation of this paper. Pedrycz [lo] showed the entropy change when the interval size of the support is changed. He considered the fuzzy set with triangular membership function. In this paper, the result of [lo] is extended to the other types of fuzzy sets. Next, we consider three common types of fuzzy numbers through arithmetic operations. These operations include addition, subtraction and multiplication. The resultant fuzzy number after arithmetic operations has its entropy. What is the relationship between the entropies of the resultant fuzzy number and the original fuzzy numbers is the main task of this paper. For the triangular fuzzy numbers, these results has been proposed in [l 11. However, this paper extends the consideration to the other three common types of fuzzy numbers. This paper is organized into five sections. In Section 2, we introduce some properties of the entropy. In Section 3, we discuss the relationship of the entropies between the fuzzy number after arithmetic operations and the original fuzzy numbers. Some examples are illustrated in Section 4. Finally, a brief conclusion is included in Section 5.
2. Preliminary A fuzzy set A is defined in a universal set X, where X is real and finite. Let A(x): x + [0, l] be the membership function of the fuzzy set A for x E X. Let a measure of fuzziness be denoted by H(A) and have the following properties [2]: 1. H(A) = 0, if A is a crisp set in X. 2. H(A) is a unique maximum if A(x) = *, Vx E X. 3. For two fuzzy sets A1 and AZ, if A,(x) < A,(x) for A,(x) < f and A,(x) > A,(x) for Al(x) 2 i then H(A1) 2 H(&). 4. H(A”) = H(A), where A” is the standard complement of A, i.e., A”(x) = 1 - A(x). As mentioned in [2] the measure of fuzziness H(A) can be regarded as an “entropy” of the fuzzy set A. At a fixed element x,
WA(x))= W(x)),
(2.1)
where the entropy function h: [O, l] + [0, l] is monotonically increasing in [O,i] and monotonically decreasing in (i, 11; moreover, h(u) = 0, as u = 0 and 1; and h(u) = 1, as u = 3. Some well-known entropy functions are shown in the following: if u E [0, 31, 2u 2(1 - u) if UE [$, 11,
I
(2.2)
h(u) p 4u(l - u).
(2.3)
h(u) p - u lnu - (1 - u)ln(l - u),
(2.4)
h(u) p
and
where (2.4) is called Shannon’s function [15]. To determine a global entropy H(A) of the fuzzy set A independent of x, we can integrate over the universal set X as follows [lo]: H(A) 4 s XSX
h(A(x))&)dx,
(2.5)
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445
where p(x) is the probability density function of the available data in X. It is known that the larger H(A) is, the more is the fuzziness of the fuzzy set A. Now, we will show that the global entropy H(A) depends on the size of the support of the fuzzy set A. For the membership function of the fuzzy set A(x), A =denotes the or-cut of A, i.e. A ~ ~- {xlA(x) >>.~, x eX}; A"" is the strong or-cut of A, i.e., A ~+ ~- {xlA(x) > ~, x e S }. If e = 0, A °+ is called the "support" of A. Definition 1. Suppose we have any two fuzzy sets A1 and A2 with the support A °+ = (al, bl) A°+ = (a2, b2) = X, respectively. If A1 (~) = A2(~),
c
X and
(2.6)
where = a l + c1,~ =a2 +(Ca/(bl - al)) (b2 - a2), 2 e [ax, bl] and )~e [a2, b2]; or
Ax(Y) = A2(~),
(2.7)
where x = a l - 4 - C l , £ = b 2 - ( C l / ( b l - a l ) ) ( b 2 - a 2 ) , x 6 [ a l , b l ] and £ e [ a 2 , b2]. Then we call A1 and A2 to be the "same type of fuzzy sets". (a) and (b) of Fig. 1 illustrate the forms (2.6) and (2.7), respectively. It is also noted that if b2 = - al, a2 = - bl then let A2 be called the "image" of A1 and be denoted by Ai(see Fig. 2), where A1 and A2 also satisfy (2.7). o+ where the "size" of A °+ is defined as It is obvious that the size of (A~-)°+ is equal to the size of A1, bl - a l .
~'~
bl
f
0
aa
b2
22,
(a)
a~
~,~
b~
a2
Co)
Fig. 1.
b2
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al
b~
~-
~
0t
Fig. 2.
L e m m a 1 (Chen and W a n g [1]). For the above same type of f u z z y sets A 1 and A 2 satisfying (2.6), suppose p(x) = s, where s is a constant over [al, b l ] u [ a 2 , b2], then H(A~) = kH(a2),
(2.8)
where k = (bl - al)/(b2 - a2) and H(Ai) z~ Sb, p(x)h(ai(x))dx, i = 1, 2. R e m a r k 1. L e m m a 1 states that ifAx is "fatter" (or "thinner") than e n t r o p y of A~ is k times of the e n t r o p y of A2.
A 2
with the ratio k > 1 (or k < 1) then the
R e m a r k 2. This l e m m a does not constrain that Ai, i = 1, 2, must be convex, piecewise linear, n o r m a l or a specific type. F r o m L e m m a 1, we can get the following result easily.
Theorem 1. For a f u z z y set A and its image A - , we have (2.9)
H(A) = H ( A - ) .
Proof. F o r the two fuzzy sets A and A-, where the support A °÷ = (a, b) c X and ( A - ) °+ = ( - b, - a) c X, we have A(2) = A - (~), where ~ = a + k, ~ = - a - k. Thus, we can use the same way of the p r o o f of L e m m a 1 to prove the result (2.9). [] Corollary 1. Suppose two f u z z y sets Ax and A 2 satisfy (2.7); then (2.10)
H(A1) = kH(A2), where k and H(Ai), i = 1, 2, are defined in Lemma 1.
Proof. Since T h e o r e m 1 says that H ( A 1 ) = H(A?). By L e m m a 1 and (2.7), we have H ( A ? ) = kH(A2). Therefore, it is obvious that (2.11) holds. [] R e m a r k 3. F o r the aforementioned two fuzzy sets A 1 and A 2 satisfying (2.6) or (2.7), if b 2 it does not need that al = a2 and bl = b2, the global e n t r o p y H(A1) is equal to H(A2).
a 2 =- b l -
al,
but
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447
3. The entropy change through arithmetic operations Let us recall the notations of arithmetic operations on fuzzy numbers. A fuzzy set A is called a "fuzzy number" if it is convex and normal [6]. Suppose A, B are two fuzzy numbers; we have [8]
(A®B)~ z~ A ~ ® B ~, as(0, 1];
(3.1)
moreover,
xCA ~x.~A~'
~A(x)~{O
Then by the decomposition theorem [8], we have
A®B~- U ~(A®B),
(3.2)
~C0, 1]
where ® denotes any arithmetic operation which may be addition, subtraction, multiplication, and division. 0 denotes the standard fuzzy union. We consider here three basic arithmetic operations: addition (i.e. A + B), subtraction (i.e. A - B), and a simple multiplication (i.e. k" A where k is a constant). It is known that if the fuzzy numbers A and B are triangular forms then A + B, A - B, and k. A, k ~ R, are also triangular fuzzy numbers [6]. For a set of triangular fuzzy numbers Ai, i = 1, 2, ..., n, and Bj,j = 1, 2 . . . . . m, the entropies of H(~iAi®ZjBj), Y.iH(Ai), and y, jH(B~) have been studied in [11]. Here, we try to study the entropy variation for the other types of fuzzy numbers with arithmetic operations. Let us consider several fuzzy numbers with different type of membership function as follows [8]: 1
Type (i): Al(X) =
+ pl(x - r) 2'
x~A~, (3.3a)
elsewhere,
Type (ii): A2(x) =
{~
-Ip2(x-r)r xeA~2, elsewhere,
(3.3b)
where e is a very small fixed constant, i.e. 1 >> e > 0. Moreover, A~=
r-
+ ~/
Pl
Type (iii): Aa(x)=
~
- Pl 2
in(3.3a) and
A~=[r+(lne/p2),r-(lne/p2)]
in(3.3b).
_1
i 1 11
x ~ r - --., r + P3 elsewhere,
, (3.3c)
where r e R for which the membership grade is required to be one; Pi > 0, for i = 1, 2, 3, is a parameter to determine the rate at which, for each x, the function decreases with the increasing difference Ir - x l. Remark 4. In order to give a finite size of the supports (bounded support) of Type (i) and Type (ii) fuzzy numbers, we have to define Ai(x) = 0 as x¢A~ i = 1, 2, in (3.3a) and (3.3b).
W.-J. Wung, C.-H. Chiu / Fuzzy Sets and Systems 103 (1999) 443-455
(a) A,(x) in (3.3a) with ~2 and e = 0.027 .(b) A,(x)in (3.3b) with ~2 and E = 0.05.
(c) As(x
(3.3~) with r=2. Fig. 3.
Next, we. will show that A and B are the fuzzy numbers with the same type of membership function then A + B, A - B, and k. A, kE R, are also the same type as A and B. For the fuzzy numbers of Type (i), A,(x) = l/(1 + pI(x - r#) and AI(X) = l/(1 + p~(x - rz)‘); then by (3.1) and (3.2) we have
1 (A, + AI)(x) = i l0
1 + -$(x - (rl +
XE (A, + Ail)&,
elsewhere;
1 (A, - A&(x) =
(3.4a)
r2V
1 + -$(x - (rI 0
XE(A, - A$, (3.4b)
r2))’
elsewhere;
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449
and x e (kA1) ~, el
(3.4c)
1 + - ~ (x - - krl) 2
kAl(x) =
elsewhere;
0
where q --&~
+ v/Up2. Moreover, by a simple calculation, in (3.4a)
(A1+,41)'=
(rl+r2)-
1 +
e,(rl+r2)+
+ / x/
=
--
--
d- ~/P2)
N/
/3
(rl
-- r2) +
-I-~p2f~l~
e
I
1
in (3.4b); and in (3.4c)
It is seen that A1 + ,41, A1 - - 4 1 and k ' A l are the same type as A1 and .4~. The detail derivation for (3.4a)-(3.4c) is shown in the appendix. For the fuzzy numbers of Type (ii) or Type (iii), we also can have the similar derivation for them to get the same type A~ + Ai and k.A~, i = 2, 3. For a set of fuzzy numbers Ai, i = 1, 2 . . . . . n, and Bj, j = 1, 2, ..., m, next will discuss the entropies H ( ~ A ~ ® ~.jBj), ~ H ( A ~ ) , and 2 j H ( B j ) . 3.1. Addition
First, let ® be the addition " + " and let the support A°+= (a,, b~) c X. Suppose
C = ~, A~.
(3.5)
i=l
Note: For the fuzzy numbers of Type (i) and Type (ii), A °+ = A~+, i --- 1, 2.
Here, the membership function of Ai, i = 1, 2, ..., n belongs to the one of the three Types (i), (ii), and (iii), n n C o+ = (c, d) c X, where c = Yi= 1 a~ and d = y~= 1 bi. From (3.4a) and appendix, it is easy to see C and A~, Vi, are same type then C(£) : A,(~), i = 1, 2, ..., n, where k
= ai + k, £ = c + 7 - - - -
oi -- ai
(d - c).
Therefore, the following result is given.
(3.6)
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450
Theorem 2. For the above f u z z y numbers C and As in (3.5), suppose p(x) = s, where s is a constant over X , we have
(3.7)
H(C) = ~" H(A,). i=1
where H(" ) is defined in (2.5).
Proof. From (3.6), we have Ai(x)=C
)
c +~(x-ai)
=C(y),xe[ai,
bi],i= l,2, ...,n,
where y = c +
d-c
bi - ai
(x - ai).
Since d y / d x = (d - c)/(bi - ai), then tt(A1) + H(A9
-
=
bl - al c
+ ... + I-I(A.) =
s'h(C(y))dy-+
fabl s . h ( A l ( x ) ) d x 1 b2 - a2 c
+
fab2s . h ( A 2 ( x ) ) d x 2
s.h(C(y))dy+
+ ... +
f:" s . h ( A n ( x ) ) d x n
bn - an ;a ... "4-----Z--~cd~d s'h(C(y))dy
(bl + b2 + ... + bn) - (al + a2 + ... + an) ( e Jc s" h(C(y)) dy d-c
= fcd s" h(C(y)) dy = H(C).
From (3.5), this implies that
i=1
i=1
3.2. Subtraction
Next, let ® denote the subtraction" - ". Suppose we have two fuzzy numbers B~ and B2 on the same type (either Type (i), Type (ii) or Type (iii)). From the decomposition theorem, BI - B2 = B1 + BE still holds for the above three types of fuzzy numbers. Therefore, the following result is obvious. Theorem 3. For the above two f u z z y numbers B1 and B2, suppose p(x) = s = constant over X; we have H ( B , - B2) = H ( B , ) + H(BE).
Proof. Since B, - B2 = B~ + B~. By Theorems 2 and 1 H(B,
-
BE) = H ( B , + B~) = H ( B O + H ( B ~ ) = H(B1) + H(B2).
[]
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451
Corollary 3. For aforementioned fuzzy numbers Ai, i = 1, 2, ..., n, in (3.4), we have H(A~ -- A2 . . . . .
(3.8)
A~) = ~ H(Ak), k=l
H(i~
Ai - j ~
(3.9)
AJ) = k=l~ H(Ak)
where I1 and I2 are two crisp sets of digits and I 1 u I 2
=
{1, 2, ... ,n}, Ilf312 = O.
Proof. By Theorems 2 and 3, it is easy to prove (3.8) and (3.9).
[]
3.3. Multiplication of a fuzzy number by a constant Now, we like to discuss the entropy ofk" A, where k is a constant and A is a fuzzy number belonging to any one of the above three types. Here, the scale k can be considered as a fuzzy number K that has the membership value K(x) = 1 as x = k and the membership value is zero for x ~ k. Thus, k" A can be seen as a multiplication between two fuzzy numbers K and A.
Theorem 4. For a constant k and a fuzzy number A as above, we have H(k.A) = k.H(A).
Proof. Suppose the support A °+ = (a, b) c X and C = k" A then it follows C O+= (c, d) c X, where c = ka and d = kb. Moreover C(kx) = A(x). Since c - d = k(b - a), by Lemma 1, it is easy to get H(C) = k'H(A). That means we have H(k.A) = k.H(A). [] Remark 5. F r o m the above analysis, it is seen that the fuzziness increases if several fuzzy numbers go through the addition and subtraction operations.
Remark 6. It is known that the considered fuzzy numbers here are all symmetric (see Fig. 3). If we consider triangular fuzzy numbers which may be not symmetric. Their entropy relation between the resultant fuzzy number and original fuzzy numbers through arithmetic operations needs further discussion [11]. Ref. [11] shows that, in fact, Theorems 2, 3 and 4 are also applicable to triangular fuzzy numbers.
4. Three examples Example 1. Consider two fuzzy numbers A and B of Type (i) as follows:
A(x) =
B(x) =
{1
+ (X -- 3) 2
{ 1
x ~ A °'°1 = [3 -- V/-~, 3 + V / ~ ] , elsewhere;
+ 4(x - 5) 2 elsewhere.
452
W.J.
Wang, C.-H. Chiu /Fuzzy
Sets and System
103 (1999) 443-455
By arithmetic operation, we have 1
x+t+B)O.O’=
1 +;(x - 8)’
(A + B)(x) =
elsewhere; 1
x E (A + B)O.O’=
-2+@9,
1 + ;(x + 2)2
(A - B)(x) =
0
1
elsewhere; 1
(2/I)(x) =
-2-;JG,
x E(~A)O.‘~= [6 - 2,/%, 6 + 2 @I,
1 +;(x - 6)2 0
elsewhere.
Let us take the entropy function as (2.3) and the entropy as (2.5) with p(x) = s, where s is a constant over X. Then, we get H(A) x 5.484s, H(B) z 2.742s, H(A + B) x 8.226s, H(A - B) x 8.226s, H(2A) z 10.968s. According to above result, we know that H(A + B) = H(A - B) = H(A) + H(B) and H(2A) = 2H(A). Example 2. Consider two fuzzy numbers A and B of Type (ii) as follows: exp-lx-31
A(x) =
o
B(x) =
;
-M*
xeAO.O’ = [3 + ln(O.Ol), 3 - ln(O.Ol)], elsewhere;
- 5)l
x E
B”.” = [5 + iln(O.Ol), 5 - ~ln(O.Ol)],
elsewhere.
Then, we have A + B, A - B, and 2A as follows: exp-IWK-88)l
(A + B)(x) =
i
o
XE(~ + qO.01 = [8 + $ln(O.Ol), 8 - $ln(O.Ol)],
elsewhere; = [ - 2 + jln(O.Ol), - 2 - +ln(O.Ol)],
(A _ B)(x) =
(2AJ(xl =
~-“1’2”‘-6”
;l;JQc,“’
= [6 + 2 ln(O.Ol), 6 - 2 ln(O.Ol)],
Suppose the entropy function (2.3) is chosen and by (2.5) with p(x) = s, where s is a constant over X. We can get H(A) x 3.92s, H(B) x 1.86s, H(A + B) w 5.48s, H(A - B) x 5.48s and H(2A) x 7.84s. So, it is obviously H(A + B) = H(A - B) = H(A) + H(B) and H(2A) = 2H(A).
W.-J. Wang, C.-H. Chiu / Fuzzy Sets and Systems 103 (1999) 443-455
Example 3.
453
We have two fuzzy numbers A and B as Type (iii) follows:
[01+ cos((x - 3)~)]
A(x) =
2
x e [-2, 4], elsewhere;
B(x) =
i1 + cos(2(x - 5)n)] 2 elsewhere.
A + B, A - B, and 2. A are given as follows:
(A + B)(x) =
[1 + cos(~(x - 8)n)] 2 elsewhere,
0
[1 + cos(~(x + 2)7t)] (A - B)(x) =
2
-
½],
elsewhere;
0
2-A(x) =
xeE -},
_[01+ cos(½(x - 6)n)] 2
x e [4, 8], elsewhere.
Let us choose the entropy function as (2.3) and the entropy as (2.5) with p(x) = s, where s denote a constant over X. We get H ( A ) = s, H ( B ) = ½s, H(A + B ) = H ( A - B ) = ~s and H ( 2 A ) = 2s. It is obvious that H(A + B) = H(A - B) = H(A) + H(B) and H(2A) = 2H(A).
5. Conclusion This paper has presented several results about the entropy variation of fuzzy numbers. First, the entropies of two same type of fuzzy sets have a constant gain relationship. The gain depends on the sizes of their supports. Second, a fuzzy set and its image have the same entropy. Furthermore, for any one type (i), (ii) or (iii) fuzzy numbers, after some arithmetic operations, their type is kept and the entropy of the resultant fuzzy number can be obtained by calculating the individual entropy of the original fuzzy numbers.
Appendix Here, we prove that if the fuzzy numbers A~ and ,4~, i = 1, 2, 3, are one of the above three types then At + Ai, A~ - Ai and k . & , k e R , are also same type as Ax and A1. First, we consider the fuzzy numbers of Type (i). According to arithmatic operations of intervals, we have
[X//1--~
1~-]1--~ ~.~=E~//1-~
~-I1--~
454
W.-J. Wang, C.-H. Chiu / Fuzzy Sets and Systems 103 (1999) 443-455
So, we can get (A1 +'41)~ --- [ ( r l + r2)
--
(X/~I
q-
~2)~l-ct
, (ri
-4- r 2 ) -b
(A1 A1)a I(rl r2)( 1 (kA1)~= krl
-k
1-
Pl~
,kr~ +
( X / ~ 1 q- . / --i- - ~ . 1-~ -] ~/P2,/~/ ~
;
_[
1 - -
~] Pl ¢z A"
By decomposition theorem, we obtain
{
(A1 + A1)(x) =
1
xe(A1 + At) ~,
+ 1 (x - (rl + r2))2 elsewhere;
xe(A1 (A 1 _ 21)(x) --
-
~Zli)e ,
-4- 1 (X -- (rl -- r2)) 2
elsewhere; and
kA1 (x) =
10+ Pl-~ (x 1
krl) 2
x e (kA 1)~,
elsewhere; where q ~- ~/1/pl + 1V/~2. It is easy to see that Al + -41, Ai - .4~, and k ' A 1 are the same type as A1 and A1. Easily, Type (it) and Type (iii) have the same conclusion by the similar derivation.
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