On additions of interactive fuzzy numbers

Transcription

1 On additions o interactive uzzy numbers Christer Carlsson IAMSR, Åbo Akademi University, Lemminkäinengatan 14B, FIN Åbo, Finland christer.carlsson@abo.i Robert Fullér Department o Operations Research, Eötvös Loránd University, Pázmány Péter sétány 1C, H-1117 Budapest, Hungary ruller@mail.abo.i Abstract: In this paper we will summarize some properties o the extended addition operator on uzzy numbers, where the interactivity relation between uzzy numbers is given by their joint possibility distribution. 1 Introduction A uzzy number A is a uzzy set o the real line R with a normal, uzzy convex and continuous membership unction o bounded support. Any uzzy number can be described with the ollowing membership unction, ( ) a t L i t [a α, a] α 1 i t [a, b],a b, A(t) = ( ) t b R i t [b, b + β] β 0 otherwise where [a, b] is the peak o A; a and b are the lower and upper modal values; L and R are shape unctions: [0, 1] [0, 1], with L(0) = R(0) = 1 and L(1) = R(1) = 0 which are non-increasing, continuous mappings. We shall call these uzzy numbers o LR-type and use the notation ã =(a, b, α, β) LR. I R(x) =L(x) =1 x, we denote ã =(a, b, α, β). The amily o uzzy numbers will be denoted by F. Aγ-level set o a uzzy number A is deined by [A] γ = {t R A(t) γ}, iγ > 0 and [A] γ =cl{t R A(t) > 0} (the closure o the support o A) iγ =0. An n-dimensional possibility distribution C is a uzzy set in R n with a normalized membership unction o bounded support. The amily o n-dimensional possibility distribution will be denoted by F n.

2 Let us recall the concept and some basic properties o joint possibility distribution introduced in [27]. I A 1,...,A n Fare uzzy numbers, then C F n is said to be their joint possibility distribution i A i (x i ) = max{c(x 1,...,x n ) x j R,j i}, holds or all x i R, i =1,...,n. Furthermore, A i is called the i-th marginal possibility distribution o C. For example, i C denotes the joint possibility distribution o A 1, A 2 F, then C satisies the relationships max C(x 1,y)=A 1 (x 1 ), y max C(y, x 2 )=A 2 (x 2 ), y or all x 1,x 2 R. Fuzzy numbers A 1,...,A n are said to be non-interactive i their joint possibility distribution C satisies the relationship C(x 1,...,x n ) = min{a 1 (x 1 ),...,A n (x n )}, or all x =(x 1,...,x n ) R n. A unction T :[0, 1] [0, 1] [0, 1] is said to be a triangular norm (t-norm or short) i T is symmetric, associative, non-decreasing in each argument, and T (x, 1) = x or all x [0, 1]. Recall that a t-norm T is Archimedean i T is continuous and T (x, x) <xor all x ]0, 1[. Every Archimedean t-norm T is representable by a continuous and decreasing unction :[0, 1] [0, ] with (1) = 0 and T (x, y) = [ 1] ((x)+(y)) where [ 1] is the pseudo-inverse o, deined by { [ 1] 1 (y) i y [0,(0)] (y) = 0 otherwise The unction is the additive generator o T. Let T 1, T 2 be t-norms. We say that T 1 is weaker than T 2 (and write T 1 T 2 )it 1 (x, y) T 2 (x, y) or each x, y [0, 1]. The basic t-norms are (i) the minimum: min(a, b) = min{a, b}; (ii) Łukasiewicz: T L (a, b) = max{a + b 1, 0}; (iii) the product: T P (a, b) =ab; (iv) the weak: { min{a, b} i max{a, b} =1 T W (a, b) = 0 otherwise (v) Hamacher [7]: and (vi) Yager T H γ (a, b) = ab γ +(1 γ)(a + b ab),γ 0 T Y p (a, b) =1 min{1, p [(1 a) p +(1 b) p ]}, p>0. Using the concept o joint possibility distribution we introduced the ollowing extension principle in [3].

3 Deinition 1.1. [3] Let C be the joint possibility distribution o (marginal possibility distributions) A 1,...,A n F, and let : R n R be a continuous unction. Then will be deined by C (A 1,...,A n )(y) = C (A 1,...,A n ) F, sup C(x 1,...,x n ). (1) y=(x 1,...,x n) We have the ollowing lemma, which can be interpreted as a generalization o Nguyen s theorem [25]. Lemma 1. [3] Let A 1,A 2 Fbe uzzy numbers, let C be their joint possibility distribution, and let : R n R be a continuous unction. Then, [ C (A 1,...,A n )] γ = ([C] γ ), or all γ [0, 1]. Furthermore, C (A 1,...,A n ) is always a uzzy number. Let C be the joint possibility distribution o (marginal possibility distributions) A 1,A 2 F, and let (x 1,x 2 )=x 1 + x 2 be the addition operator. Then A 1 + A 2 is deined by (A 1 + A 2 )(y) = sup y=x 1+x 2 C(x 1,x 2 ). (2) I A 1 and A 2 are non-interactive, that is, their joint possibility distribution is deined by C(x 1,x 2 ) = min{a 1 (x 1 ),A 2 (x 2 )}, then (2) turns into the extended addition operator introduced by Zadeh in 1965 [26], (A 1 + A 2 )(y) = sup y=x 1+x 2 min{a 1 (x 1 ),A 2 (x 2 )}. Furthermore, i C(x 1,x 2 )=T (A 1 (x 1 ),A 2 (x 2 )), where T is a t-norm then we get the t-norm-based extension principle, (A 1 + A 2 )(y) = sup y=x 1+x 2 T (A 1 (x 1 ),A 2 (x 2 )). (3) For example, i A 1 and A 2 are uzzy numbers, T is the product t-norm then the supproduct extended sum o A 1 and A 2 is deined by (A 1 + A 2 )(y) = sup A 1 (x 1 )A 2 (x 2 ), x 1+x 2=y and the sup-t H γ extended addition o A 1 and A 2 is deined by (A 1 + A 2 )(y) = sup x 1+x 2=y A 1 (x 1 )A 2 (x 2 ) γ +(1 γ)(a 1 (x 1 )+A 2 (x 2 ) A 1 (x 1 )A 2 (x 2 )).

4 I T is an Archimedean t-norm and ã 1, ã 2 Fthen their T -sum can be written in the orm A 2 := ã 1 +ã 2 A 2 (z) = [ 1] ((ã 1 (x 1 )) + (ã 2 (x 2 ))),z R, where is the additive generator o T. By the associativity o T, the membership unction o the T -sum A n := ã 1 + +ã n can be written as A n (z) = sup x 1+ +x n=z [ 1] ( n i=1 ) (ã i (x i )),z R. Since is continuous and decreasing, [ 1] is also continuous and non-increasing, we have ( n ) A n (z) = [ 1] in (ã i (x i )),z R. x 1+ +x n=z 2 Additions o interactive uzzy numbers Dubois and Prade published their seminal paper on additions o interactive uzzy numbers in 1981 [4]. Since then the properties o additions o interactive uzzy numbers, when their joint possibility distribution is deined by a t-norm have been extensively studied in the literature [1-3, 5-24]. In 1992 Fullér and Keresztalvi [6] generalized and extended the results presented in [4]. Namely, they determined the exact membership unction o the t-norm-based sum o uzzy intervals, in the case o Archimedean t-norm having strictly convex additive generator unction and uzzy intervals with concave shape unctions. They proved the ollowing theorem, Theorem 2.1. [6] Let T be an Archimedean t-norm with additive generator and let ã i =(a i,b i,α,β) LR, i =1,...,n, be uzzy numbers o LR-type. I L and R are twice dierentiable, concave unctions, and is twice dierentiable, strictly convex unction then the membership unction o the T -sum Ãn =ã 1 + +ã n is 1 i A n z B n ( ( ))) An [ 1] z (n L i A n nα z A n nα Ã n (z) = ( ( ))) z Bn (n [ 1] R i B n z B n + nβ nβ 0 otherwise i=1 where A n = a a n and B n = b b n.

5 The results o Theorem 2.1 have been extended to wider classes o uzzy numbers and shape unctions by many authors. In 1994 Hong and Hwang [8] provided an upper bound or the membership unction o T -sum o LR-uzzy numbers with dierent spreads. They proved the ollowing theorem, Theorem 2.2. [8] Let T be an Archimedean t-norm with additive generator and let ã i =(a i,α i,β i ) LR, i =1, 2, be uzzy numbers o LR-type. I L and R are concave unctions, and is a convex unction then the membership unction o the T -sum à 2 =ã 1 +ã 2 is less than or equal to A 2(z) = ( ( (2 [ 1] L 1/2+ (A ))) 2 z) α i A 2 α 1 α 2 z A 2 α (2α ( ( ))) A2 (2 [ 1] z L 2α i A 2 α z A 2 ( ( ))) z A2 (2 [ 1] R 2β i A 2 z A 2 + β ( ( (2 [ 1] R 1/2+ (z A ))) 2) β i A 2 + β z A 2 + β 1 + β 2 2β 0 otherwise where β = max{β 1,β 2 }, β = min{β 1,β 2 }, α = max{α 1,α 2 }, α = min{α 1,α 2 } and A 2 = a 1 + a 2. The In 1995 Hong [9] proved that Theorem 2.1 remains valid or concave shape unctions and convex additive t-norm generator. In 1996 Mesiar [22] showed that Theorem 2.1 remains valid i both L and R are convex unctions. In 1997 Mesiar [23] generaized Theorem 2.1 to the case o nilpotent t-norms (nilpotent t-norms are nonstrict continuous Archimedean t-norms). In 1997 Hong and Hwang [11] gave upper and lower bounds o T -sums o LR-uzzy numbers ã i =(a i,α i,β i ) LR, i =1,...,n, with dierent spreads where T is an Archimedean t-norm. They proved the ollowing two theorems, Theorem 2.3. [11] Let T be an Archimedean t-norm with additive generator and let ã i =(a i,α i,β i ) LR, i =1,...,n, be uzzy numbers o LR-type. I L and R are concvex unctions, then the membership unction o their T -sum Ãn =ã 1 + +ã n

6 is less than or equal to ( ( ))) 1 (n [ 1] L n I L (A n z) A ( ( ))) n(z) = 1 (n [ 1] R n I R (z A n ) 0 otherwise, where I L (z) = in and I R (z) = in { x1 α x n α n { x1 β x n β n i A n n i=1 α i z A n i A n z A n + n i=1 β i } x x n = z, 0 x i α i,i=1,...,n, } x x n = z, 0 x i β i,i=1,...,n. Theorem 2.4. [11] Let T be an Archimedean t-norm with additive generator and let ã i =(a i,α i,β i ) LR, i =1,...,n, be uzzy numbers o LR-type. Then à n (z) A n (z) = ( ( ))) An (n [ 1] z L i A n (α α n ) z A n α α n ( ( ))) An (n [ 1] z R i A n z A n +(β β n ) β β n 0 otherwise, In 1997, generalizing Theorem 2.1, Hwang and Hong [15] studied the membership unction o the t-norm-based sum o uzzy numbers on Banach spaces and they presented the membership unction o inite (or ininite) sum (deined by the sup-t-norm convolution) o uzzy numbers on Banach spaces, in the case o Archimedean t-norm having convex additive generator unction and uzzy numbers with concave shape unction. In 1998 Hwang, Hwang and An [16] approximated the strict triangular norm-based addition o uzzy intervals o L-R type with any let and right spreadss. In 2001 Hong [12] showed a simple method o computing T -sum o uzzy intervals having the same results as the sum o uzzy intervals based on the weakest t-norm T W. 2.1 Shape preserving arithmetic operations Shape preserving arithmetic operations o LR-uzzy intervals allow one to control the resulting spread. In practical computation, it is natural to require the preservation o

7 the shape o uzzy intervals during addition and multiplication. Hong [13] showed that T W, the weakest t-norm, is the only t-norm T that induces a shape-preserving multiplication o LR-uzzy intervals. In 1995 Kolesarova [19, 20] Kolesarova proved the ollowing theorem, Theorem 2.5. (a) Let T be an arbitrary t-norm weaker than or equal to the Łukasiewicz t-norm T L ; T (x, y) T L (x, y) = max(0,x+ y 1), x, y [0, 1]. Then the addition based on T coincides on linear uzzy intervals with the addition based on the weakest t-norm T W ; i.e., (a 1,b 1,α 1,β 1 ) (a 2,b 2,α 2,β 2 )= (a 1 + a 2,b 1 + b 2, max(α 1,α 2 ), max(β 1,β 2 )). (b) Let T be a continuous Archimedean t-norm with convex additive generator. Then the addition based on T preserves the linearity o uzzy intervals i and only i the t-norm T is a member o Yager s amily o nilpotent t-norms with parameter p [1, ), T = Tp Y, and (x) =(1 x) p. Then T1 Y = T L and or p (0, ), (a 1,b 1,α 1,β 1 ) (a 2,b 2,α 2,β 2 )= (a 1 + a 2,b 1 + b 2, (α q 1 + αq 2 )1/q, (β q 1 + βq 2 )1/q ), where 1/p +1/q =1, i.e. q = p/(p 1). In 1997 Mesiar [24] studied the triangular norm-based additions preserving the LRshape o LR-uzzy intervals and conjectured that the only t-norm-based additions preserving the linearity o uzzy intervals are those described in Theorem 2.5. He proved the ollowing theorem, Theorem 2.6. [24] Let a continuous t-norm T be not weaker than or equal to T L (i.e., there are some x, y [0, 1] so that T (x, y) >x+ y 1 > 0). Let the addition based on T preserve the linearity o uzzy intervals. Then either T is the strongest t-norm, T = T M,orT is a nilpotent t-norm. In 2002 Hong [14] proved Mesiar s conjecture. Theorem 2.7. [14] Let a continuous t-norm T be not weaker than or equal to T L. Then the addition based on T preserves the linearity o uzzy intervals i and only i the t-norm T is either T M or a member o Yagers amily o nilpotent t-norms with parameter p (1, ), T = T Y p, and (x) =(1 x) p. 2.2 Additions o interactive uzzy numbers Until now we have summarized some properties o the addition operator on interactive uzzy numbers, when their joint possibility distribution is deined by a t-norm. It is clear that in (3) the joint possibility distribution is deined directly and pointwise

8 rom the membership values o its marginal possibility distributions by an aggregation operator. However, the interactivity relation between uzzy numbers may be given by a more general joint possibility distribution, which can not be directly deined rom the membership values o its marginal possibility distributions by any aggregation operator. Drawing heavily on [3] we will now consider some properties o the addition operator on completely correlated uzzy numbers, where the interactivity relation is given by their joint possibility distribution. Let C be a joint possibility distribution with marginal possibility distributions A and B, and let (x 1,x 2 )=x 1 + x 2, the addition operator in R 2. In [3] we introduced the notation, A + C B = C (A, B). Deinition 2.1. [5] Fuzzy numbers A and B are said to be completely correlated, i there exist q, r R, q 0such that their joint possibility distribution is deined by C(x 1,x 2 )=A(x 1 ) χ {qx1+r=x 2}(x 1,x 2 )=B(x 2 ) χ {qx1+r=x 2}(x 1,x 2 ), (4) where χ {qx1+r=x 2}, stands or the characteristic unction o the line In this case we have, {(x 1,x 2 ) R 2 qx 1 + r = x 2 }. [C] γ = { (x, qx + r) R 2 x =(1 t)a1 (γ)+ta 2 (γ),t [0, 1] } where [A] γ =[a 1 (γ),a 2 (γ)]; and [B] γ = q[a] γ + r, or any γ [0, 1]. Deinition 2.2. [5] Fuzzy numbers A and B are said to be completely positively (negatively) correlated, i q is positive (negative) in (4). Now let us consider the extended addition o two completely correlated uzzy numbers A and B, (A + C B)(y) = sup C(x 1,x 2 ). y=x 1+x 2 That is, (A + C B)(y) = sup A(x 1 ) χ {qx1+r=x 2}(x 1,x 2 ). y=x 1+x 2 Then rom (2) and (4) we ind, [A + C B] γ =(q + 1)[A] γ + r, (5) or all γ [0, 1]. IA and B are completely negatively correlated with q = 1, that is, [B] γ = [A] γ + r, or all γ [0, 1], then A + C B will be a crisp number. Really, rom (5) we get [A + C B] γ =0 [A] γ + r = r, or all γ [0, 1].

9 Figure 1: Completely negatively correlated uzzy numbers with q = 1. That is, the interactive sum, A + C B, o two completely negatively correlated uzzy numbers A and B with q = 1 and r =0, i.e. A(x) =B( x), x R, will be (crisp) zero. On the other hand, a γ-level set o their non-interactive sum, A + B, can be computed as, [A + B] γ =[a 1 (γ) a 2 (γ),a 2 (γ) a 1 (γ)], which is a uzzy number. In this case (i.e. when q = 1)anyγ-level set o C are included by a certain level set o the addition operator, namely, the relationship, [C] γ {(x 1,x 2 ) R x 1 + x 2 = r}, holds or any γ [0, 1] (see Fig. 1). On the other hand, i q 1 then the uzziness o A + C B is preserved, since [A + C B] γ =(q + 1)[A] γ + r constant, or all γ [0, 1] and y R. (see Fig. 2). Really, in this case the set {(x 1,x 2 ) [C] γ x 1 + x 2 = y} consists o a single point at most or any γ [0, 1] and y R.

10 Figure 2: Completely negatively correlated uzzy numbers with q 1. Note 2.1. The interactive sum o two completely negatively correlated uzzy numbers A and B with A(x) =B( x) or all x R will be (crisp) zero. 3 Summary In this paper we have summarized some properties o the addition operator on interactive uzzy numbers, when their joint possibility distribution is deined by a t-norm or by a more general type o joint possibility distribution. Reerences [1] B. De Baets and A. Markova, Addition o LR-uzzy intervals based on a continuous t-norm, in: Proceedings o IPMU 96 Conerence, (July 1-5, 1996, Granada, Spain), [2] B. De Baets and A. Marková-Stupňanová, Analytical expressions or addition o uzzy intervals, Fuzzy Sets and Systems, 91(1997) [3] C. Carlsson, R. Fullér and P. Majlender, Additions o Completely Correlated Fuzzy Numbers, in: FUZZY IEEE 2004 CD-ROM Conerence Proceedings, Budapest, July 26-29, 2004,

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