Crisp Profile Symmetric Decomposition of Fuzzy Numbers

Applied Mathematical Sciences, Vol. 10, 016, no. 8, 1373-1389 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ams.016.59598 Crisp Profile Symmetric Decomposition of Fuzzy Numbers Maria Letizia Guerra 1 Department of Mathematics University of Bologna, Italy Luciano Stefanini Department of Economics, Society and Politics University of Urbino, Italy Copyright c 015 Maria Letizia Guerra and Luciano Stefanini. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The present paper investigates the crisp profile symmetric decomposition in order to analyze its properties when working with fuzzy arithmetic operations and shape-preservation (symmetry). We are also interested in the definition of generalized differentiability based on the mentioned decomposition. 1 Introduction Many approaches have been introduced to define fuzzy arithmetic operations satisfying certain desired properties that may not always be satisfied in the classical extension principle approach (see [17]) or its approximations (see [14]): shape preservation (e.g. [4], [5], [14]), reduction of the overestimation effect (e.g. [8], [9]), requisite constraints (e.g. [11], [10]) and distributivity of multiplication and division (e.g. [], [1], [13]). These problems are essentially approached by joining representations of fuzzy quantities and fuzzy operations (e.g. [3], [7], [8], [11]). 1 Corresponding author

1374 Maria Letizia Guerra and Luciano Stefanini In this paper, we refine the crisp profile symmetric (hereafter CPS) decomposition of the fuzzy numbers into three additive components that can be thought as a crisp part, a symmetric fuzzy part and a profile of symmetry. The decomposition (firstly introduced in [16]) contributes to define some properties of fuzzy calculus in a rigorous way. The decomposition is also used to suggest some approximations of fuzzy operations that reduce the range of fuzziness (with respect to the classical exact operations) and assure distributivity. Basic notions In the a cut setting, a fuzzy number can be identified as follows: Definition 1 A fuzzy number (or interval) u is any pair (u, u + ) of functions u ± : [0, 1] R satisfying the following conditions: (i) u : α u α R is a bounded monotonic increasing (non decreasing) function α [0, 1] ; (ii) u + : α u + α R is a bounded monotonic decreasing (non increasing) function α [0, 1] ; (iii) u α u + α α [0, 1]. We denote the (α = 1) cut [u 1 < u + 1 ] by [û, û + ]; if û < û + we have a fuzzy interval and if û = û + we have a fuzzy number. The notation [u] α = [u α, u + α ] denotes explicitly the α cuts of u and u and u + are the lower and upper branches on u. We denote by F I the set of fuzzy intervals and by F F I the set of fuzzy numbers. If u α = û and u + α = û +, α we have a crisp interval or a crisp number; we denote by F I and by F the corresponding sets. If û = û + = 0 we obtain a 0 fuzzy number and denote the corresponding set by F 0. If u α + u + α = û + + û, α then the fuzzy interval is called symmetric; we denote by S I and by S the sets of symmetric fuzzy intervals and numbers; S 0 = S F 0 will be the set of symmetric 0 fuzzy numbers. The fuzzy number u is positive if u α > 0, α [0, 1] and it is negative if u + α < 0, α [0, 1]; the sets of positive and negative fuzzy numbers are denoted by F + and F respectively and their symmetric subsets are denoted by S + and S. Given two fuzzy intervals u = (u, u + ) and v = (v, v + ) then the standard arithmetic operations are defined.

Crisp profile symmetric decomposition of fuzzy numbers 1375 3 The CPS representation In the unidimensional case, given a fuzzy number u F with α cuts [u] α = [u α, u + α ] it is possible to define the following quantities: û = [ û, û +] I (1) corresponding to the core [ ] u 1, u + 1 obtained as the (α = 1) cut, that is a standard compact real interval; ũ α = u α + u + α û + û + corresponding to the symmetry profile of u where ũ : [0, 1] R is a given function satisfying ũ 1 = 0 when α = 1 and P is the set of all profile functions; u α = u+ α u α û+ û corresponding to the symmetric fuzzy component of u where u is a symmetric fuzzy number with core given by the singleton {0} and S 0 is the family of all such fuzzy numbers. In the proposed framework the α cuts of a fuzzy number can be written as: { u + α = û + + ũ α + u α u α = û (4) + ũ α u α () (3) implying that any fuzzy number can be decomposed in terms of a triplet u = (û, ũ, u) = I P S 0 (5) and defining the CPS representation where û I is the crisp number or interval, ũ P is the crisp symmetry profile and u S 0 is the 0 symmetric fuzzy number. By use of decomposition (5) we can characterize the fuzzy intervals or numbers as follows: 1. u F 0 û = û + = 0. u F û + + ũ α u α, α 3. u F + û + ũ α u α, α 4. u S ũ α = 0, α 5. u S 0 û = û + = 0, ũ α = 0, α

1376 Maria Letizia Guerra and Luciano Stefanini 6. u S û u α, ũ α = 0, α 7. u S + û + u α, ũ α = 0, α Note also that ũ 1 = 0 and u α ũ α u α α [0, 1] and that the profile function induces a linear operator; in fact, if u = (û, ũ, u) and v = ( v, ṽ, v) then u + v = (û + v, ũ + ṽ, u + v) and, k R, ku = (kû, kũ, k u) and ũ + v = ũ + ṽ, ku = kũ. Let s denote by P the set of all possible profile functions: Theorem Any ũ : [0, 1] R with ũ 1 = 0. u = (û, ũ, u) F I P S 0 represents a fuzzy interval if and only if the following condition is satisfied by the pair (ũ, u) P S 0 : α < α = ũ α ũ α u α u α (6) Proof. If u is a fuzzy interval with û, ũ and u defined by (1), () and (3) respectively, then (6) is immediate. Suppose now that (6) is valid, then for α = α < 1 and α = 1 we have ũ α u α and necessary condition is satisfied. By definition (4) we then have: u α u + α α( as u α 0). For α < α the following is true and u + α = û + + ũ α + u α û + + ũ α + u α = u + α u α = û + ũ α u α û + ũ α u α = u α so that u + α is decreasing (not increasing) and u α is increasing (not decreasing). It follows that u is a proper fuzzy interval. Definition 3 (valid pair (ũ, u)) We say that a pair (ũ, u) P S 0 is a valid pair if it satisfies condition (6), i.e. if ũ + u F 0 is a fuzzy number.

Crisp profile symmetric decomposition of fuzzy numbers 1377 In terms of the above definition, we can say that any u = (û, ũ, u) F I P S 0 represents a fuzzy interval if and only if (ũ, u) is a valid pair. On the other hand, valid pairs of P S 0 are the elements of F 0, the 0-fuzzy numbers. Note that if ũ and u are differentiable functions with respect to α ]0, 1[, then condition (6) can be stated in terms of first derivatives as ũ α u α α ]0, 1[. Furthermore, as ũ 1 = u 1 = 0 and the absolute local variations of ũ are not greater then local variations of u, it follows that if (ũ, u) P S 0 is a valid pair then also ũ α u α, α [0, 1]. Using decomposition u = (û, ũ, u) with û F I, ũ P and u S 0 and supposing that (ũ, u) is a valid pair, then some possibilities arise and some of them are of interest: (û, 0, 0) is crisp; (0, 0, u) is in S 0 ; (û, 0, u) is in S I ; (0, ũ, u) is in F 0 ; (û, ũ, u) is in F I ; (0, ũ, 0) is in P. Also a configuration (û, ũ, 0) with û = [û, û + ] may be of interest as it represents an interval of fixed length but of varying position, depending on the profile function ũ : at different degree of possibility α, the position of the fixed length interval changes by following profile ũ. If ϕ u : R [0, 1] is the membership function of fuzzy interval u, of the form 0 for x u 0 u l (x) for x [ u 0, û ] ϕ u (x) = 1 for x [û, û + u r (x) for x [ ] û +, u + 0 0 for x u + 0 with u l (x) increasing and u r (x) decreasing, we have that α [0, 1] : { ϕu (u α ) = α = u l (u α ) ϕ u (u + α ) = α = u r (u + α ) In terms of decomposition (4) we can write the following relations, α ]0, 1[ : { u l (x) = α x = û + ũ α u α u r (x) = α x = û + + ũ α + u α The fuzziness of u is essentially contained in component u S 0, while the profile function ũ is related to the asymmetry of u with respect to crisp component û. Starting with one or more valid pairs, it is not difficult to construct other valid pairs. For example, if (ũ, u) is given we can consider any function F : P S 0 P S 0 such that F (ũ, u) = ( F (ũ, u), F (ũ, u)) is a valid pair with F : P S 0 P

1378 Maria Letizia Guerra and Luciano Stefanini and F : P S 0 S 0. A particular case is related to the extension of a given function f : R R to a 0-fuzzy number u = (0, ũ, u) F 0 for which f(u) = min{f(x) x [u] α} + max{f(x) x [u] α } f(u) = max{f(x) x [u] α} min{f(x) x [u] α } f(0) f(0). Similarly, starting with n valid pairs (ũ i, u i ), i = 1,,..., n, and setting ũ = (ũ 1,..., ũ n ) P n, u = (u 1,..., u n ) (S 0 ) n we can construct two functions F : P n (S 0 ) n P and F : P n (S 0 ) n S 0 such that ( F (ũ, u), F (ũ, u)) is a valid pair. A useful result is given by the following Proposition 4 Let (ũ, u), (ṽ, v) P S 0 be two valid pairs. Then, a, b, c, d, e R also ( aũ + bṽ + cũṽ + dũv + euṽ, a u + b v + ( c + d + e )uv ) is a valid pair. Proof. Consider α < α [0, 1]; we have the following inequalities (remember that u α u α 0 and v α v α 0): a(ũ α ũ α ) + b(ṽ α ṽ α )+ +c(ũ α ṽ α ũ α ṽ α )+ +d(ũ α v α ũ α v α )+ +e(u α ṽ α u α ṽ α ) a ũ α ũ α + b ṽ α ṽ α + + c [ ũ α ṽ α ṽ α + ṽ α u α u α ] + + d [ ũ α v α v α + v α u α u α ] + + e [u α ṽ α ṽ α + ṽ α u α u α ] a (u α u α ) + b (v α [ v α )+ ] + c u α ( v α }{{} v α ) + v α (u α u α }{{} ) + [ ] + d u α ( v α }{{} v α ) + v α (u α u α }{{} ) + [ ] + e u α ( v α }{{} v α ) + v α (u α u α }{{} ) a (u α u α ) + b (v α v α ) + ( c + d + e )(u α v α u α v α ). The following results hold, too.

Crisp profile symmetric decomposition of fuzzy numbers 1379 Proposition 5 (adapted from [1]) If (ũ, u) and (ṽ, v) are two valid pairs, then also ( w, w) with is a valid pair. w α = min{ũ α u α, ṽ α v α } + max{ũ α + u α, ṽ α + v α } w α = max{ũ α + u α, ṽ α + v α } min{ũ α u α, ṽ α v α } Finally, the Hausdorff distance on F I is defined by: D (u, v) = sup α [0,1] { { max u α vα, u + α v α + }} (7) If we use the decomposition { u α = û + ũ α u α u + α = û + + ũ α + u α (and similarly for v), then { u α v α = (û v ) + (ũ α ṽ α ) + (v α u α ) u + α v + α = (û + v + ) + (ũ α ṽ α ) + (u α v α ) ; we can obtain a modified distance on F I by considering the three components D (u, v) = max { û v, û + v + } D (u, v) = sup ũ α ṽ α α [0,1] D (u, v) = sup u α v α α [0,1] and by defining D (u, v) = D (u, v) + D (u, v) + D (u, v). (8) It is possible to see that D (u, v) is equivalent to D (u, v) as D D, D D, D D and then D D 5D. 4 Basic fuzzy arithmetic with CPS decomposition In [6] a comparison index for interval and fuzzy numbers ordering is defined. It is based on the essential notion of generalized Hukuhara difference, introduced

1380 Maria Letizia Guerra and Luciano Stefanini by several authors with different denominations and here approached following [15]: it is defined for two compact intervals A = [a, a + ] and B = [b, b + ] by where A gh B = [ min { a b, a + b +}, max { a b, a + b +}] = (â b; a b ). â = a+ + a, a = a+ a describe the so called midpoint-radius representation A = (â; a). The gh-difference satisfies several properties: 1. A gh A = {0} ;. (i) (A + B) gh B = A; (ii)a gh (A B) = B; (iii)a gh (A+B) = B; 3. A gh B exists if and only if B gh A and ( B) gh ( A) exist and A gh B = ( B) gh ( A) = (B gh A); 4. In general, B A = A B does not imply A = B; but A gh B = B gh A = C if and only if C = C and C = {0} if and only if A = B; 5. If B gh A exists then either A + (B gh A) = B or B (B gh A) = A and both equalities hold if and only if B gh A is a singleton set; 6. If B gh A = C exists, then for all D either (B + D) gh A = C + D or B gh (A + D) = C D. Several properties may be showed when the fuzzy numbers are represented through the CPS decomposition. Proposition 6 Given u = (û, ũ, u) F and v = ( v, ṽ, v) F then the ghdifference w = u g v is a fuzzy number w = (ŵ, w, w) F that satisfies the following properties: ŵ = û g v, w α = ũ α ṽ α, u α v α if u v S 0 w α = v α u α if v u S 0 and u g v exists if and only if one of the two conditions is satisfied: 1) u v S 0, û g v exists in case (i) and (ũ ṽ, u v) V (P, S 0 ) or 1) v u S 0, û g v exists in case (ii) and (ũ ṽ, v u) V (P, S 0 ). Proof. Is detailed in [15].

Crisp profile symmetric decomposition of fuzzy numbers 1381 5 The gh-derivative in the CPS framework In [1] the gh-derivative in terms of the CPS decomposition of fuzzy numbers is analyzed. Given a fuzzy-valued function f : [a, b] R F with level-cuts [f (x)] α = [f α (x), f + α (x)], the CPS representation decomposes f(x) in terms of the following three additive components: f (x) = f (x) + f (x) + f (x) where f [ (x) = f (x), f (x)] + is a crisp interval-valued function, f (x) = { fα (x) α [0, 1]} is a family of real valued profile functions f α : [a, b] R [ and f ](x) is a fuzzy valued function f α : [a, b] R F of 0-symmetric type f (x) = [ f α α (x), f α (x) ] ; the three components are defined as follows f (x) = [f (x)] 1, i.e. f (x) = f 1 (x) and f + (x) = f + 1 (x) f α (x) = f α (x) + f + α (x) f 1 (x) + f + 1 (x) for all α [0, 1] f α (x) = f α + (x) f 1 + (x) f α (x) f1 (x) 0 for all α [0, 1] and they satisfies the following equations: [ [f (x)] α = f (x), f (x)] + + f α (x) + [ f α (x), f α (x) ], i.e. (9) f α (x) = f (x) + f α (x) f α (x) f + α (x) = f + (x) + f α (x) + f α (x) Equation (9) defines the CPS decomposition of f (x) R F and the symmetry of the fuzzy number holds if and only if f α (x) = 0 for all α [0, 1], so that we can call f (x) + f (x) the symmetric part of f (x). Let s now assume that the lower and the upper functions fα and, f α + are differentiable w.r.t. x for all α; then also f and f + are differentiable w.r.t. x, and f α + f α are differentiable w.r.t. x for all α. Necessarily the following equations hold: ( ) ( ) f ( ) ( ) α (x) = f (x) + fα (x) f α (x) ( ) ( ) f + ( ) ( ) α (x) = f + (x) + fα (x) f α (x)

138 Maria Letizia Guerra and Luciano Stefanini and the level cuts of the gh-derivative of f are given by: [ ] ( ) [ { f + ( ) = ( ) gh fα min α f ( ) } { ( ) f α, f + ( ) ( ) + f α, max f ( ) }] ( ) f α, f + + f α Some interesting facts can be deduced from (10). In particular the monotonicity conditions can be expressed in terms of the CPS decomposition as follows: (10) fα (x) x α f α (x) x α 0 and fα (x) x α OR + f α (x) x α 0 α fα (x) x α f α (x) x α 0 and fα (x) x α + f α (x) x α 0 α In particular, for a symmetric fuzzy number f(x) with f 1 (x) = f + 1 (x) we obtain gh-differentiability if and only if ( f α ) (x) is monotonic w.r.t. α. 6 Properties of arithmetic operations Standard arithmetic operations between fuzzy numbers can be handled in terms of the CPS decomposition where (11) is equivalent to u = (û, ũ, u) F P S 0 (11) { u + α = û + ũ α + u α u α = û + ũ α u α α [0, 1]. (1) Addition, scalar multiplication and subtraction are immediate. Let u = (û, ũ, u) and v = ( v, ṽ, v) be two fuzzy numbers, then we can write: u + v = (û + v, ũ + ṽ, u + v) ; (13) ku = (kû, kũ, k u), for k R (14) u v = u + ( v) = (û v, ũ ṽ, u + v). (15)

Crisp profile symmetric decomposition of fuzzy numbers 1383 It may be of interest to note that addition and difference have the same 0- symmetric fuzzy components u + v and they differ (only) in the crisp and the profile parts. On the other hand, it is well known that: and that u = (0, 0, u) S 0 u = u u, v S 0 u + v = u v = v u = u v. When we consider the multiplication of u, v F with u = (û, ũ, u) and v = ( v, ṽ, v) that are fuzzy numbers with { { u + α = û + ũ α + u α v + u and α = v + ṽ α + v α α = û + ũ α u α vα = v + ṽ α v α The α cuts of the fuzzy product uv are: [uv] α = [ ] [ ] u α, u + α v α, v α + = [ min { } (û + ũ α ) vα u α v α, (û + ũα ) v α + u α v + α, max { (û + ũ α ) vα + u α vα, (û + ũ α ) v α + + u α v α + }]. (16) The CPS decomposition is useful to measure the effect of the multiplication on the symmetry of the product fuzzy number. It is immediate to see that if u, v S 0 i.e. u = (0, 0, u) and v = (0, 0, v), then uv = (0, 0, uv) S 0 is the exact product. If we take u, v F 0 i.e. u = (0, ũ, u) and v = (0, ṽ, v) then the α cuts of the product uv F 0 are given by [uv] α = [ũ α ṽ α u α v α u α ṽ α ũ α v α, (17) ũ α ṽ α + u α v α + u α ṽ α + ũ α v α ]; it follows that the elements of the decomposition of uv are: and (uv) α = ũ α ṽ α + u αṽ α + ũ α v α u α ṽ α ũ α v α = ũ α ṽ α + max{u α ṽ α, ũ α v α } (uv) α = u α v α + u αṽ α ũ α v α + u α ṽ α + ũ α v α = u α v α + w α. where w α {u α ṽ α, ũ α v α, u α ṽ α, ũ α v α }

1384 Maria Letizia Guerra and Luciano Stefanini and we see how the profile functions of u and v contribute to the definition of the product. We also see from (17) that S 0 absorbs F 0 in the sense that if u S 0 and v F 0 then uv = (0, 0, uv + uṽ ) S 0 (in a similar way, F 0 absorbs F as if u F 0, v F then uv F 0 ). Consider now the case u, v S i.e. u = (û, 0, u) and v = ( v, 0, v). Then the product uv is given (after some algebra) by [uv] α = û v + [min{u α v α ûv α + u α v, u α v α ûv α u α v }, (18) max{u α v α + ûv α + u α v, u α v α + ûv α u α v }]. Note that, in this case, min{.,.} 0 and max{.,.} 0 and if min{.,.} = u α v α ûv α + u α v then max{.,.} = u α v α + ûv α + u α v. maxα + minα maxα minα If we compute (ũv) α = and (uv) α = we obtain the following three cases: Case 1. min{.,.} = u α v α ûv α +u α v and max{.,.} = u α v α + ûv α +u α v : then (ũv) α = u α v α and (uv) α = ûv α + u α v and a positive asymmetry is introduced by the operation; Case. min{.,.} = u α v α ûv α u α v and max{.,.} = u α v α + ûv α u α v : then (ũv) α = u α v α and (uv) α = ûv α +u α v and a negative asymmetry is introduced; Case 3. min{.,.} = u α v α ûv α u α v and max{.,.} = u α v α + ûv α + u α v : then (ũv) α = ûv α + u α v ûv α u α v and (uv) α = u α v α + 1( ûv α + u α v + ûv α + u α v ) and an unsigned asymmetry is introduced. In order to define the fourth arithmetic operation, we introduce the reciprocal of z that is the fuzzy number given by z 1 with α cuts [ ] 1 [z 1 1 ] α =, then the division is defined as u z = uz 1. Note that if u = (0, 0, u α ) S 0 then u S z 0 as the α cuts are [ u z ] α = [ u α max { 1 z + α, 1 z α z + α z α }, u α max { 1 z + α, 1 z α }]. If z = (ẑ, 0, z) S S + is symmetric, then it is easy to see that z 1 = (ẑ 1, z 1, z 1 ) is not symmetric as z 1 α = z 1 α = z α ẑ(ẑ z α) z α ẑ z. α

Crisp profile symmetric decomposition of fuzzy numbers 1385 In [16] we investigate when the distributivity of multiplication and division is satisfied. 7 Examples In Multiplication 1, the two fuzzy numbers u and v are both trapezoidal with linear left and right branches u =, 0, 1, and v =, 1, 0,. In Multiplication. we use u and v in the decomposed form with û = 1, ũ α = (1 α) /, u α = (1 α)( α)/ and v =, ṽ α = ũ α, v α = 1.5u α. Figure 1: Multiplication 1 In Division 1, the two fuzzy numbers u and v are both trapezoidal with linear left and right branches u = 1,, 3, 4 and v =, 3, 5, 7. In Division, the two fuzzy numbers u and v are both trapezoidal with linear left and right branches u = 1, 0, 0, 5 and v = 1,,, 3. Note that u F 0. In the calculus of fuzzy expressions we take an example with triangular u = 0, 1, and v = 1,, 3 and compute the Klir ([10]) operation ( uv ) u+v E compared to the approximation App = (u v) : (u + v). Finally, we compare the multiplication with the arithmetic introduced by Ma et al. in [1]. In our notation, the input fuzzy numbers of their last example are u = (1, α(1 α), α(α+1) ) and v = (, α(α 1), α(α+1) ); their multiplication produces the linear symmetric ( uv = (, 0, 1 α) having α cuts [uv] α = ( ) ( ) ) [1 + α, 3 α] while u v =, α(α 1), α(α+1).

1386 Maria Letizia Guerra and Luciano Stefanini Figure : Multiplication Figure 3: Division 1

Crisp profile symmetric decomposition of fuzzy numbers 1387 Figure 4: Division Figure 5: Klir example

1388 Maria Letizia Guerra and Luciano Stefanini 8 Conclusions Figure 6: Ma et al. example We study properties of the gh-derivative and of fuzzy arithmetic operations that can be expressed in terms of the Crisp Profile Symmetric decomposition and we show that the introduced representation enlarges the fuzzy numbers features. References [1] B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, 30 (013), 119-141. http://dx.doi.org/10.1016/j.fss.01.10.003 [] P. Benvenuti, R. Mesiar, Pseudo-arithmetical operations as a basis for the general measure and integration theory, Information Sciences, 160 (004), 1-11. http://dx.doi.org/10.1016/j.ins.003.07.005 [3] M. Delgado, M. A. Vila, W. Voxman, On a canonical representation of fuzzy nuhmbers, Fuzzy Sets and Systems, 93 (1998), 15-135. http://dx.doi.org/10.1016/s0165-0114(96)00144-3 [4] D. Dubois, H. Prade, Fuzzy numbers, an overview, Chapter in Analysis of Fuzzy Information (Mathematics), J. C. Bezdek (ed.), CRC Press, 1988, 3-39.

Crisp profile symmetric decomposition of fuzzy numbers 1389 [5] M.L. Guerra, L. Stefanini, Approximate Fuzzy Arithmetic Operations Using Monotonic Interpolations, Fuzzy Sets and Systems, 150 (005), no. 1, 5-33. http://dx.doi.org/10.1016/j.fss.004.06.007 [6] M.L. Guerra, L. Stefanini, A comparison index for interval ordering based on generalized Hukuhara difference, Soft Computing, 16 ( 01), no. 11, 1931-1943. http://dx.doi.org/10.1007/s00500-01-0866-9 [7] A. Kandel, Fuzzy Mathematical Techniques with Applications, Addison Wesley, Reading, Mass, 000. [8] A. Kaufman M. M. Gupta, Introduction to Fuzzy Arithmetic, Theory and Applications, Van Nostrand Reinhold Company, New York, 1985. [9] G. J. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice Hall, 1995. [10] G. J. Klir, Fuzzy arithmetic with requisite constraints, Fuzzy Sets and Systems, 91 (1997), 165-175. http://dx.doi.org/10.1016/s0165-0114(97)00138-3 [11] G. J. Klir, Uncertainty Analysis in Engineering and Sciences, Kluwer, 1997. [1] M. Ma, M. Friedman, A. Kandel, A new fuzzy arithmetic, Fuzzy Sets and Systems, 108 (1999), 83-90. http://dx.doi.org/10.1016/s0165-0114(97)00310- [13] R. Mesiar, J. Rybarik, Pan-operations structure, Fuzzy Sets and Systems, 74 (1995), 365-369. http://dx.doi.org/10.1016/0165-0114(94)00314-w [14] L. Stefanini, L. Sorini, M.L. Guerra, Parametric Representations of Fuzzy Numbers and Application to Fuzzy Calculus, Fuzzy Sets and Systems, 157 (006), 43-455. http://dx.doi.org/10.1016/j.fss.006.0.00 [15] L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets and Systems, 161 (010), no. 11, 1564-1584. http://dx.doi.org/10.1016/j.fss.009.06.009 [16] L. Stefanini, M.L. Guerra, On fuzzy arithmetic operations: some properties and distributive approximations, International Journal of Applied Mathematics, 19 (006), 171-199. [17] L. A. Zadeh, Fuzzy Sets, Information and Control, 8 (1965), 338-353. Received: October 10, 015; Published: April 9, 016