Parametric representation of fuzzy numbers and ... - Semantic Scholar

Fuzzy Sets and Systems 157 (2006) 2423 – 2455 www.elsevier.com/locate/fss

Parametric representation of fuzzy numbers and application to fuzzy calculus Luciano Stefaninia , Laerte Sorinia , Maria Letizia Guerraa, b,∗ a Faculty of Economics, University of Urbino “Carlo Bo”, Italy b Department MATEMATES, University of Bologna, Italy

Received 26 March 2005; received in revised form 1 February 2006; accepted 14 February 2006 Available online 10 March 2006

Abstract We present several models to obtain simple parametric representations of the fuzzy numbers or intervals, based on the use of piecewise monotonic functions of different forms. The representations have the advantage of allowing flexible and easy to control shapes of the fuzzy numbers (we use the standard
-cuts setting, but also the membership functions are obtained immediately) and can be used directly to obtain error-controlled-approximations of the fuzzy calculus in terms of a finite set of parameters. The general setting is the Hermite-type interpolation, where the values and the slopes of the monotonic interpolators are given by appropriate parameters and the overall errors of the fuzzy computations can be controlled within a prefixed tolerance by eventually augmenting the total number of pieces (and of the parameters) by which the results are obtained. The representations are designed to model the lower and the upper extremal values of each
-cut (compact) intervals of the fuzzy numbers and are able to produce almost any possible configuration (differentiable, continuous or with a finite number of discontinuity points) by using parametric monotonic functions of different types. We show applications in the standard fuzzy calculus and we stress the generality and the applicability of the proposed representation to a large class of problems, including the numerical solution of fuzzy differential equations, the fuzzy linear regression and the stochastic extensions of the fuzzy mathematics. The proposed model is called the Lower–Upper representation and we denote the associated fuzzy numbers or intervals as LU-fuzzy. © 2006 Elsevier B.V. All rights reserved. Keywords: Parametric fuzzy numbers; Fuzzy calculus; Fuzzy arithmetic; Fuzzy extension principle; Monotonic splines

1. Introduction Twenty five years ago, after the forty years-old introduction of the fuzzy sets by Zadeh [60], Dubois and Prade (see [15]) stated the exact analytical fuzzy mathematics and introduced the well known LR model and the corresponding formulas for the fuzzy operations (see also [18]). More recently, the literature on fuzzy numbers has grown in terms of contributions to the fuzzy arithmetic operations and to the use of simple formulas to approximate them; an extensive recent survey and bibliography on fuzzy intervals is in [19]. ∗ Corresponding author.

E-mail addresses: [email protected] (L. Stefanini), [email protected] (L. Sorini), [email protected] (M.L. Guerra). 0165-0114/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2006.02.002

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In general, the arithmetic operations on fuzzy numbers can be approached either by the direct use of the membership function (by the Zadeh extension principle) or by the equivalent use of the
-cuts representation (introduced by Goetschel and Voxman in [25]). In [30] Guerra and Stefanini suggested the use of monotonic splines to approximate (or to model directly) fuzzy numbers, using diverse piecewise interpolation forms (cubic, rational and mixed cubic-exponential) and derived a procedure to decide the number and locations of the nodes so that the error of the approximation of product and ratio of fuzzy numbers is controlled by possible insertion of additional nodes into the piecewise interpolation. The simplicity of computations is maintained due to the adoption of local algorithms that are guaranteed to produce monotonic interpolants, as is necessary for a valid representation of the
-cuts. By this approach, it is possible to define a parametric representation of fuzzy numbers that allows a large variety of possible shapes (types of membership functions) and is very simple to implement, with the advantage of obtaining easily a much wider set than for standard LR model. As the representation is designed to model the lower and upper extremal values of the
-cuts, we give it the name LU-representation and we denote the associated fuzzy numbers and intervals as LU-fuzzy. An additional important advantage of the proposed parametric representation is that it produces a subspace of (the complete space of) the fuzzy numbers and the results of fuzzy calculus can be parametrized by the same forms, by calculating the corresponding values; in this sense, the representation is closed with respect to the operations, within an error tolerance that can be controlled by eventually refining the representation within the same class. Differently with respect to the standard approach, that is mainly focused on approximating the membership function (see [6]), requiring different discretizations for the support of each fuzzy number involved, the use of the
-cuts representation has the relevant advantage of being applied to the same [0, 1] interval for all the fuzzy numbers involved in the computations (see [31,32,39] for a recent analogous approach). Furthermore, the high quality of monotonic splines (or other piecewise approximators that preserve monotonicity) is obtained even with a small number of nodes; as reported in [30], from 3 to 5 nodes are sufficient to obtain errors of the order of 0.1% and in this paper we will show that up to 5 nodes give good results also in massive calculations. An extended overview of the techniques of monotonic interpolation (polynomial, rational or based on other forms) is summed up in the works by Kocic and Milovanovic [42] and Wolberg and Alfy [56], where both local and global procedures are described that preserve monotonicity and/or convexity of the shapes. The almost general idea to obtain monotonic interpolants is by introducing parametrized first derivatives (giving the slopes) of the modelling functions and by estimating the parameters so that the derivatives are of the required sign. In many cases, a global estimation procedure, based on some optimization criterion, is required to obtain the final monotonic interpolant; this is the case, for example, of the pure-cubic spline for which the positivity of the first derivatives at the nodes is not sufficient to obtain the global monotonicity (see [56] for the details). The situation changes when the model function is not restricted to the simple cubic polynomials and either rational functions or higher degree polynomials or exponential functions (and possibly other) are allowed: in this cases, in fact, it is possible to control global monotonicity by the use of some additional parameters and in most cases by a single value, easy to be determined. Diverse types of monotonic rational interpolations have been presented in different papers (that we will describe and refer later) and a mixed cubic-exponential Hermite-type interpolator has been recently introduced by Stefanini in [54]. In this paper, we will present different monotonic function models that have all the advantage of a very simple calculation with the direct use (without the need of re-estimation) of the parameters that produce monotonicity. The organization of the paper is the following: Section 2 contains a brief introduction to the fuzzy numbers and the notations that we will use and to the fundamental elements of the fuzzy operations and calculus; Section 3 introduces several models for the monotonic interpolation that will be used to obtain the parametrized LU-representation; in Section 4 we detail the LU-representation of the fuzzy numbers and some of its properties and use in the standard fuzzy arithmetic operations. We also see the basic relationship between the LU and the LR models. In Section 5 we describe the use of the LU-fuzzy numbers in diverse applications of fuzzy mathematics, such as calculation of ranking functions, computation of fuzzy valued functions, integration and differentiation of fuzzy valued functions, numerical solution of fuzzy differential equations and solution of fuzzy linear systems and fuzzy linear regression. We conclude the paper by a brief discussion of further study to improve the computational performance of the representation and to extend the use of LU-fuzzy numbers in other applications.

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2. Basic fuzzy calculus In what follows, according to the representation theorem for fuzzy numbers or intervals (see [25]), we use the so called
-cut setting to define a fuzzy number or interval; a fuzzy number (or interval) u is completely determined by any pair u = (u− , u+ ) of functions u± : [0, 1] −→ R, defining the end-points of the
-cuts, satisfying the three conditions: (i) u− :
−→ u−
∈ R is a bounded monotonic increasing (nondecreasing) left-continuous function ∀
∈]0, 1] and right-continuous for
= 0; (ii) u+ :
−→ u+
∈ R is a bounded monotonic decreasing (nonincreasing) left-continuous function ∀
∈]0, 1] and right continuous for
= 0; + (iii) u−

u
∀
∈ [0, 1].

+ − + If u− 1 < u1 we have a fuzzy interval and if u1 = u1 we have a fuzzy number; for simplicity we refer to fuzzy numbers as intervals. + We will then consider fuzzy numbers of normal, upper semicontinuous form and we assume that the support [u− 0 , u0 ] of u is compact (closed and bounded). The notation + u
= [u−
, u
],


∈ [0, 1]

denotes explicitly the
-cuts of u. We refer to u− and u+ as the lower and upper branches on u, respectively. A trapezoidal fuzzy number is denoted by u = a, b, c, d and has
-cuts u
= [a +
(b − a), d −
(d − c)]; a triangular fuzzy number is denoted by u = a, b, c and has
-cuts u
= [a +
(b − a), c −
(c − b)]. The arithmetic operations for two fuzzy numbers u = (u− , u+ ) and v = (v − , v + ) are defined in the standard way, in terms of the
-cuts for
∈ [0, 1]: − + + Addition: (u + v)
= [u−
+ v
, u
+ v
]; Scalar Multiplication: for given k ∈ R, + − + (ku)
= [min{ku−
, ku
}, max{ku
, ku
}]; + + − Subtraction: (u − v)
= [u−
− v
, u
− v
]; − + Multiplication: (uv)
= [(uv)
, (uv)
] where, − − − + + − + + (uv)−
= min{u
v
, u
v
, u
v
, u
v
}, − − − + + − + + (uv)+
= max{u
v
, u
v
, u
v
, u
v
}, u + Division: if 0 ∈ / [v0− , v0+ ], ( uv )
= [( uv )−
, ( v )
] where  − − + +
u − u
u
u
u
= min − , +, −, + , v
v
− v
− v
+ v
+ 
u + u
u
u
u
. = max − , , , v
v
v
+ v
− v
+

If we denote by F the set of fuzzy numbers/intervals, then the Hausdorff distance on F is defined by − + + DH (u, v) = sup {max{|u−
− v
|, |u
− v
|}}.


∈[0,1]

+ (k) Assuming regular shapes for u−
and u
, the fuzzy numbers will be of class C , k 0, with continuous derivatives up to the kth order. All models that are suggested in this paper will produce fuzzy numbers that are generally piecewise C (1) .

3. Models for monotonic interpolation Our actual main interest is in maintaining the simplicity of computations by the use of simple local monotonic approximations of the lower and upper branches of fuzzy numbers. The branches are obtained as Hermite-type interpolators

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on a finite decomposition 1 of interval [0, 1] into N subintervals: 0 =
0 <
1 < · · · <
i−1 <
i < · · · <
N = 1. Without loosing generality, we will use the same subdivision for both the lower u− (
) and upper u+ (
) branches of the fuzzy numbers involved; we can always reduce to this situation by refining two different subdivisions to their union. In each of the N subintervals Ii = [
i−1 ,
i ],

i = 1, 2, . . . , N

the values of the two functions u− (
i−1 ) = u− 0,i , u− (
i ) = u− 1,i ,

u+ (
i−1 ) = u+ 0,i ,

u+ (
i ) = u+ 1,i

and of their first derivatives (slopes) − u− (
i−1 ) = d0,i , − u− (
i ) = d1,i ,

+ u+ (
i−1 ) = d0,i ,

+ u+ (
i ) = d1,i

are assumed to be known (exact or approximated 2 ); we are interested in families of monotonic functions that satisfy the above eight Hermite-type conditions for each subinterval Ii . In general, by use of the following transformation of each subinterval Ii into the standard [0, 1] interval: t
=


−
i−1 ,
i −
i−1


∈ Ii

(1)

we can determine each piece independently and obtain general left-continuous LU-fuzzy numbers. Globally continuous or more regular C (1) fuzzy numbers can be obtained directly from the data if the following conditions are met for the values: − u− 1,i = u0,i+1 ,

+ u+ 1,i = u0,i+1 for i = 1, 2, . . . , N − 1

and possibly for the slopes − − d1,i = d0,i+1 ,

+ + d1,i = d0,i+1 for i = 1, 2, . . . , N − 1.

In this paper, following an idea presented in [30], we use simple-to-compute monotonic splines, well known and largely used in computer graphics and in geometric modelling research areas. Many other monotonic interpolators can be used; we have a preference for models that do not require any readjustment of the slopes to ensure global monotonicity, as is in general necessary to pure cubic spline and some simple fractional models (see [42] for a general discussion on monotonic or other shape preserving constraints, such as strict positivity or concavity/convexity). Let pi (t) denote the approximation of u(
) on a generic subinterval Ii of the
-decomposition, with the transformation t =
−
i−1 /
i −
i−1 , so that each subinterval is re-mapped to the standard interval [0, 1] by pi (t) = u(
i−1 + t (
i −
i−1 )), pi (t) = u (
i−1 + t (
i −
i−1 ))(
i −
i−1 ).

(2)

For simplicity of notation, we omit the subscript i and we refer to Ii as [0, 1] i.e. to the two-point Hermite interpolation problem of determining the monotonic function p(t), t ∈ [0, 1], such that p(0) = u0 ,

p (0) = d0 ,

p(1) = u1 ,

p (1) = d1 .

(3)

1 If the number of subintervals increases, then the quality of the interpolation will increase the precision of the computations. We have a preference in using a uniform subdivision of the interval [0, 1] and in refining the decomposition by successively bisecting each subinterval, producing N = 2n for n = 0, 1, 2, . . .; extended computational results (see [30]), have produced acceptable small errors, of the order of 1%, with n = 1 or 2. 2 Actually, there exist many procedures to estimate the first derivatives, given monotonic data values {
, u }N . An extensive list of methods, i i i=0 either of local or global nature, are described in [42] or in [56].

L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

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If u0
u1 (i.e. the data are increasing) then d0 0 and d1 0 are required and, if u0 u1 (i.e. the data are decreasing) then d0
0 and d1
0 are required. In particular, u0 = u1 ⇒ d0 = d1 = 0. In the following subsections, we will use the notation (m, n)-rational to mean the ratio P (t)/Q(t) of an m-degree polynomial P (t) to an n-degree polynomial Q(t). 3.1. Quadratic/quadratic rational spline Delbourgo and Gregory in [8] introduce a (2,2)-rational monotonic spline that has the following form: ⎧ ⎨ P (t) if u1 = u0 , p(t) = Q(t) ⎩u if u = u , 0

1

(4)

0

where P (t) = (u1 − u0 )u1 t 2 + (u0 d1 + u1 d0 )t (1 − t) + (u1 − u0 )u0 (1 − t)2 , Q(t) = (u1 − u0 )t 2 + (d1 + d0 )t (1 − t) + (u1 − u0 )(1 − t)2 . Without any additional parameters, the function above satisfies the Hermite interpolation conditions at the points t = 0 and t = 1. 3.2. Cubic/linear rational spline Shrivastava and Joseph in [52] introduce a (3,1)-rational monotonic spline given by p(t) =

P (t) , Q(t)

where P (t) = vu0 (1 − t)3 + wu1 t 3 + [(2v + w)u0 + vd0 ]t (1 − t)2 + [(v + 2w)u1 − wd1 ]t 2 (1 − t), Q(t) = v + (w − v)t

(5)

with v, w > 0 and w v. If v = w we obtain the ordinary cubic spline. A choice of the tension parameters v and w that guarantees the global monotonicity of p on [0, 1] is v d1 − u1 + u0  w u1 − u 0 − d 0 so that a simple choice may be, for example, v = 1 and   u1 − u0 − d0 1 w = max , 1, d1 − (u1 − u0 ) 1 −  where  is a nonnegative small number, say  ∈ [0, .1]. 3.3. Cubic/quadratic rational spline For the (3,2)-rational model there exist many forms suggested in the literature (examples are in [7,27]); the following, introduced by Gregory in [27], is of our interest as the parameters involved can be derived in a simple way P (t) p(t) = with Q(t) P (t) = u0 (1 − t)3 + (wu0 + d0 )t (1 − t)2 + (wu1 − d1 )t 2 (1 − t) + u1 t 3 , Q(t) = 1 + t (1 − t)(w − 3)

(6)

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L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

and a choice for the tension parameter w 0 to have global monotonicity is w = d0 + d1 /u1 − u0 , obtaining the ordinary cubic spline if w = 3. 3.4. Cubic/cubic rational spline The (3,3)-rational spline we work on has been proposed by Sarfraz in [50] and Sarfraz et al. in [51]: P (t) p(t) = with Q(t) P (t) = u0 (1 − t)3 + (vu0 + d0 )t (1 − t)2 + (wu1 − d1 )t 2 (1 − t) + u1 t 3 , Q(t) = (1 − t)3 + vt (1 − t)2 + wt 2 (1 − t) + t 3

(7)

with v = r(d0 + d1 /u1 − u0 ) and w = s(d0 + d1 /u1 − u0 ), r, s 1. If v = w then (7) becomes (6). 3.5. Mixed cubic-exponential interpolation The monotonic Hermite-type interpolator introduced by Stefanini in [54] is based on a mixed cubic-exponential spline; in its simpler form, it is given by 

t d0 + d1 2 p(t) = u0 + u1 − u0 − [d0 (1 − s)w + d1 s w ] ds t (3 − 2t) + 1+w 0  d0 + d1 2 d0 d1 a d0 = u0 + u1 − u0 − (8) t (3 − 2t) + − (1 − t)a + t , a a a a where a = 1 + w d0 + d1 /u1 − u0 0 to have monotonicity. We use w = d0 + d1 /u1 − u0 0 or, to work with integer exponents, w = int(d0 + d1 /u1 − u0 ). The linear case (i.e. triangular fuzzy numbers) is obtained by putting d0 = d1 = u1 − u0 and a = 3: it is easy to see that the model becomes p(t) = u0 + d0 t. If the data are quadratic, i.e. d0 + d1 = 2(u1 − u0 ), then a = 3 and the model becomes quadratic, p(t) = u0 + d0 t + (d1 − u1 + u0 )t 2 . 3.6. A parametrization of the slopes If the slopes d0 and d1 are not available, we can proceed by choosing them such that, for a given positive integer n, d0 + d1 = n(u1 − u0 ); in this case we obtain from (8) a polynomial shape. In particular, if we fix an integer n and a parameter  ∈ [0, 1], we can select (provided that u1 − u0 = 0) d0 = n(1 − )(u1 − u0 ), d1 = n(u1 − u0 )

(9)

so that d0 + d1 = n(u1 − u0 ) and a = n + 1. If  = 0 or  = 1 the model gives two extreme shapes having d0 = 0 or d1 = 0. In case (9) the spline (8) becomes

 n (u1 − u0 ) 2 3 k n+1 p(t) = u0 + . (10) (1 − t) + nt 3t − 2t + n(1 − )t n+1 k=0

If we use (9), the piecewise construction (2) with N > 1 will produce in general only a globally continuous monotonic interpolant u(
) as, at the nodes, the left and the right slopes will have different values and u(
) will not be globally C (1) . Note that if n = 2 and  = 21 we obtain the linear case and if n = 2 and  = 21 then p(t) is quadratic. In many situations it is not easy nor relevant to have the exact values of the slopes d0 , d1 ; in these cases, we can use a parametrization, analogous to (9), and write d0 = 0 (u1 − u0 ), d1 = 1 (u1 − u0 ),

0 0, 1 0

so that the different interpolators depend on the levels u0 , u1 and on the (nonnegative) parameters 0 , 1 .

(11)

L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

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Fig. 1. Standard (3,2)-rational spline.

If d0 , d1 are assigned according to (11) the models (4)–(8) assume a simplified form and it is interesting to examine the shapes according to different values of 0 and 1 . Note that, in this case, we obtain d0 + d1 = 0 + 1 0 for models (6), (7, with v = w) and (8), u1 − u 0 u1 − u0 − d0 2 − 0 − 1 w−1= −1= to be 0 for model (5); d1 − (u1 − u0 ) 1 − 1

w=

all the values of 0 0 and 1 0 are valid only for (4), (6), (7) and (8). The standard (3,2)-rational and mixed curves are illustrated in Figs. 1 and 2 for different values of (0 , 1 ). Fig. 1 describes the standard (3,2)-rational spline p(t) =

0 t (1 − t)2 + 0 t 2 (1 − t) + t 3 1 + t (1 − t)(0 + 1 − 3)

for different values of the parameters 0 and 1 and p(0) = 0, p(1) = 1, p (0) = 0 , p (1) = 1 ; the graph contains the standard curves (t, 1 − p(t)), t ∈ [0, 1] corresponding to the set of parameters (0 , 1 ) ∈ {(0, 0), (0, 2), (2, 0), (1, 1), (1, 2), (2, 1), (2, 2), (1, 5), (5, 1), (5, 5), (2, 10), (10, 2), (10, 10), (0, 20), (20, 0), (1, 20), (20, 1), (20, 20)}. Fig. 2 describes the standard mixed spline p(t) = (t 2 (3 − 2t) + 0 − 0 (1 − t)a + 1 t a )/a, where a = 1 + 0 + 1 , for different values of the parameters 0 and 1 and p(0) = 0, p(1) = 1, p  (0) = 0 , p  (1) = 1 ; the graph contains the standard curves (t, 1 − p(t)), t ∈ [0, 1] corresponding to the set of parameters (0 , 1 ) ∈ {(0, 0), (0, 2), (2, 0), (1, 1), (1, 2), (2, 1), (2, 2), (1, 5), (5, 1), (5, 5), (2, 10), (10, 2), (10, 10), (0, 20), (20, 0), (1, 20), (20, 1), (20, 20)}.

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L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

Fig. 2. Standard mixed spline.

4. LU-fuzzy numbers The monotonic models illustrated in the previous section suggest a first parametrization of fuzzy numbers obtained + by representing the lower and upper branches u−
and u
of u on the trivial decomposition of interval [0, 1], with N = 1 (without internal points) and
0 = 0,
1 = 1. In this simple case, u can be represented by a vector of 8 components − + + − − + + u = (u− 0 , u0 , u0 , u0 ; u1 , u1 , u1 , u1 ),

(12)

− − − + + + + − + where u− 0 , u0 , u1 , u1 are used for the lower branch u
, and u0 , u0 , u1 , u1 for the upper branch u
, by application of a monotonic interpolator on the whole interval
∈ [0, 1]. In particular, the slopes corresponding to u− i are denoted by u− i , etc. We will denote by z or (z) the slope associated to the element z. More generally, a parametric representation of a fuzzy number on a decomposition 0 =
0 <
1 < · · · <
N = 1 can be written as − − − + + + + u = (u− 0,i , u0,i , u1,i , u1,i ; u0,i , u0,i , u1,i , u1,i )i=1,...,N ,

with
-cuts −

−

+

+

−  +  +   u
= [pi (t
; u− 0,i , u0,i , u1,i , u1,i ), pi (t
; u0,i , u0,i , u1,i , u1,i )]i=1,2,...,N .

(13)

+ + + + − − − The functions pi (t
; u− 0,i , u0,i , u1,i , u1,i ) and pi (t
; u0,i , u0,i , u1,i , u1,i ) are obtained by the described monotonic  k,i = uk,i (
i −
i−1 ), i = 0, 1 and t
=
−
i−1 /
i −
i−1 for
∈ models using the indicated data, with u

L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

2431

− − − − − − [
i−1 ,
i ]. For N 1 we have a total of 8N parameters u− 0,1
u1,1
u0,2
u1,2
· · ·
u0,N
u1,N , uk,i 0 defining + + + + + + + the increasing lower branch u−
and u0,1 u1,1 u0,2 u1,2  · · · u0,N u1,N , uk,i
0 defining the decreasing − + + upper branch u
(obviously, also u1,N
u1,N is required). A simplification of (13) can be obtained by requiring continuous or differentiable branches; in the first case, u− 1,i = − + − − u0,i+1 and u+ = u for i = 1, 2, . . . , N − 1 while, to have differentiability, also the conditions u = u 1,i 0,i+1 k,i k,i+1 , + + uk,i = uk,i+1 are required. For the two cases we then have 6N + 2 or 4N + 4 parameters, respectively. After the LU-representation (12), it is not difficult to obtain the membership function (x) of the LU-fuzzy number u, given in general by + (x) = sup{
|x ∈ [u−
, u
]}.

(14)

In particular, corresponding to the nodes of the
-decomposition, we have + (u− i ) = (ui ) =
i

for i = 0, 1, . . . , N

and, in the differentiable case (the general piecewise differentiable case is similar),  (u− i )=

1 , u− i

 (u+ i )=

1 for i = 0, 1, . . . , N. u+ i

As reported in [30], the membership function (x) can be approximated with small errors by the use of monotonic splines similar to the ones proposed. In the following sections, we will consider only the differentiable case, for which we use the representation (we will frequently omit the
i ’s when the decomposition is fixed or it is uniform, i.e.
i = i/N , i = 0, 1, . . . , N): − + + u = (
i ; u− i , ui , ui , ui )i=0,1,...,N

or

− + + u = (u− i , ui , ui , ui )i=0,1,...,N

(15)

with the data − − + + + u− 0
u1
· · ·
uN
uN
uN−1
· · ·
u0

(16)

and the slopes u− i 0,

u+ i
0.

(17)

By the Lower–Upper representation we can define corresponding spaces of fuzzy numbers, on which the standard operations and metrics can be introduced. − − − + + + + Denote by FN = {u| u = (u− 0,i , u0,i , u1,i , u1,i ; u0,i , u0,i , u1,i , u1,i )i=1,...,N } or, in the differentiable case, − + +   FN = {u| u = (u− i , ui , ui , ui )i=0,1,...,N } the set of LU-fuzzy numbers. FN is a 4(N + 1)-dimensional space. Given two LU-fuzzy numbers − + + u = (u− i , ui , ui , ui )i=0,1,...,N

and v = (vi− , vi− , vi+ , vi+ )i=0,1,...,N

a Euclidean-like distance on  FN can be defined by dN (u, v) = u − v2N

=

u − vN 4N N

where

− 2 + + 2 − − 2 + + 2 [(u− i − vi ) + (ui − vi ) + (ui − vi ) + (ui − vi ) ].

(18)

i=0

It is worthwhile to observe that also the one-sided fuzzy numbers can be easily represented by a form similar to the LU-representation: a left-sided fuzzy number has
-cuts of the form [u−
, +∞[ and a right-sided fuzzy number

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L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

has
-cuts of the form ] − ∞, u+
] (see [28] for further details). Then, a left-sided fuzzy number can be written as − + u = (u− , u ) or a right-sided fuzzy number as u = (u+ i i i=0,1,...,N i , ui )i=0,1,...,N . 4.1. LU-fuzzy arithmetic operations The arithmetic operators associated to the LU representation can be obtained easily. The addition is defined by − − − + + + + u + v = (u− i + vi , ui + vi , ui + vi , ui + vi )i=0,1,...,N .

The scalar multiplication is defined as follows: if k 0 then − + + ku = (ku− i , kui , kui , kui )i=0,1,...,N ;

if k < 0 then + − − ku = (ku+ i , kui , kui , kui )i=0,1,...,N .

In particular, if k = −1, we have + − − −u = (−u+ i , −ui , −ui , −ui )i=0,1,...,N

and the subtraction is defined by u − v = u + (−v). We note explicitly that the scalar multiplication is always reproduced exactly in all the models for all
∈ [0, 1] but, in general, this is not true for the addition as the sum of rational or mixed functions is not always a rational or a mixed function of the same orders. A particular situation arises for addition (or subtraction) if the mixed model is used. Suppose that the two branches to (1) (1) (1) (1) (2) (2) (2) (2) be added are given by the data (u0 , u1 , u0 , u1 ) and (u0 , u1 , u0 , u1 ); the mixed model is characterized (1) (1) (1) (1) (2) (2) (2) (2) by values of w (or a ) for each data set w(1) = u0 + u1 /u1 − u0 , w(2) = u0 + u1 /u1 − u0 . If w(1)+(2) is the w parameter for the addition, then (1)

w (1)+(2) =

(1)

(2)

(2)

u0 + u1 + u0 + u1 (1)

(1)

(2)

(2)

u1 − u 0 + u 1 − u 0

and it is easy to see that w (1)+(2) is a weighted average of w (1) and w (2) as w(1)+(2) ∈ [min{w (1) , w(2) }, max{w(1) , w(2) }]; if w(1) = w(2) then it follows that w (1)+(2) = w(1) = w(2) . So, if the two fuzzy numbers to be added are modelled by a spline of the same degree, then the mixed model produces exact addition for all
∈ [0, 1]. This is true, in particular, for the fuzzy numbers having the form (10) with the same value of n in (9). In this case, in fact, if (k)

(k)

(k)

u0 = n(1 − (k) )(u1 − u0 ), then



(1) u0

(2) + u0

(1)

(2)

(1) + (2) =n 1− 2

u1 + u1 = n

(k)

(k)

 (1)

(2)

(1)

(2)

(u1 + u1 − u0 − u0 ),

(1) + (2) (1) (2) (1) (2) (u1 + u1 − u0 − u0 ) 2

so that (1)+(2) = (1) + (2) /2.

(k)

u1 = n(k) (u1 − u0 ),

k = 1, 2

L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

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For fuzzy multiplication we have an easy to implement algorithm, based on the applications of exact fuzzy multiplication at the nodes of the
-subdivision; define − − − + + − + + (uv)− i = min{ui vi , ui vi , ui vi , ui vi },

(19)

− − − + + − + + (uv)+ i = max{ui vi , ui vi , ui vi , ui vi }

(20)

and set the following: y = uv = (yi− , yi− , yi+ , yi+ )i=0,1,...,N . To implement the multiplication we can proceed as follows: let (pi− , qi− ) be the pair associated to the combination + + of superscripts + and − giving the minimum (uv)− i in (19), and similarly let (pi , qi ) the pair associated to the + combination of + and − giving the maximum (uv)i in (20), then we obtain p− q −

yi− = ui i vi i

p+ q +

and yi+ = ui i vi i ,

p− q −

p−

q−

yi− = ui i vi i + ui i vi i

p+ q +

p+

q+

and yi+ = ui i vi i + ui i vi i ,

where we use the product derivative rule to obtain the new slopes. Analogous formulas can be deduced for division z = u/v = (zi− , zi− , zi+ , zi+ )i=0,1,...,N ,

− − − + + − + + (u/v)− i = min{ui /vi , ui /vi , ui /vi , ui /vi } and

− − − + + − + + (u/v)+ i = max{ui /vi , ui /vi , ui /vi , ui /vi }.

Let (ri− , si− ) be the pair associated to the combination of + and − giving the minimum in (u/v)− i and similarly let , then it follows: (ri+ , si+ ) be the pair associated to the combination of + and − giving the maximum in (u/v)+ i r−

s−

zi− = ui i /vi i

r+

s+

and zi+ = ui i /vi i ,

r − s−

r−

s−

s−

zi− = (ui i vi i − ui i vi i )/(vi i )2

r + s+

r+

s+

s+

and zi+ = (ui i vi i − ui i vi i )/(vi i )2 .

As pointed out by the results of experimentation reported in [30], the operations above are exact at the nodes
i and have very small global errors on [0, 1]. Further, it is easy to control the error by using a sufficiently high number of nodes with max{
i −
i−1 } sufficiently small. The results in [30] have shown that both the rational (7) and the mixed (8) models perform well, with a percentage average error of the order of 0.1%. 4.2. LU-fuzzy and LR-fuzzy numbers It is well known that LR fuzzy numbers are formed by two nonincreasing shape functions L, R : [0, 1] → [0, 1] such that R(0) = L(0) = 1 and R(1) = L(1) = 0; the membership function is then ⎧  u1,L − x ⎪ ⎪ if u0,L
x
u1,L , L ⎪ ⎪ ⎪ u1,L − u0,L ⎪ ⎨ 1  if u1,L
x
u1,R , (x) = x − u 1,R ⎪ ⎪ if u1,R
x
u0,R , R ⎪ ⎪ ⎪ u0,R − u1,R ⎪ ⎩ 0 otherwise provided that u0,L
u1,L
u1,R
u0,R . The (parametric) monotonic splines can be used as models for the shape functions L and R; in fact, if d0 , d1 0 are given and we define (for example in the mixed spline model) p(t; d0 , d1 ) = 1 −

1 [3t 2 − 2t 3 + d0 − d0 (1 − t)1+d0 +d1 + d1 t 1+d0 +d1 ] 1 + d 0 + d1

(21)

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L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

Fig. 3. LU inverse approximation of an LR fuzzy number.

then p(0) = 1, p(1) = 0, p  (0) = −d0 , p  (1) = −d1 , p  (t)
0 ∀t ∈ [0, 1] and we can write the LR shapes 3 as L(t) = p(t; d0,L , d1,L ), R(t) = p(t; d0,R , d1,R ).

(22)

An LR fuzzy number can be obtained by using (21) and (22) as the shape functions with the parameters uLR = (u0,L , d0,L , u0,R , d0,R ; u1,L , d1,L , u1,R , d1,R ) provided that u0,L
u1,L
u1,R
u0,R and d0,L , d1,L , d0,R , d1,R 0. The last two pairs of parameters give the slopes d0,L , d1,L of the membership function  at the points x = u0,L and x = u1,L and d0,R , d1,R at the points x = u1,R and x = u0,R , respectively. For a given fuzzy number (22), its LU representation can be approximated with N = 1 (two points in the
-cut discretization) (see Fig. 3) − + + − − + + uLU = (u− 0 , u0 , u0 , u0 ; u1 , u1 , u1 , u1 ),

with u− 0 = u0,L , u− 1 = u1,L , u+ 0 = u0,R , u+ 1 = u1,R ,

u1,L − u0,L , d1,L u1,L − u0,L u− , 1 = d0,L u1,R − u0,R u+ , 0 = d1,R u1,R − u0,R u+ 1 = d0,R u− 0 =

(if some d0,L , d1,L , d0,R , d1,R is zero, the corresponding infinite u± slope can be assigned a BIG number). 3 Note that d = d = 1 will produce linear p(t). 0 1

(23)

L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

2435

Fig. 4. Membership functions of the fuzzy numbers uLR = (0, 0.5, 2, 0.5; 1, 0.05, 1, 0.1), vLR = (2, 0.1, 4, 0.5; 3, 0.5, 3, 0.1) and w = uv. The LR multiplication wLR = (0, 0.083, 8, 0.417; 3, 0.075, 3, 0.05) is performed by using the (LU)–(LR) relationship. All the membership functions are calculated by the mixed spline model with N = 1.

Fig. 3 represents the membership function for the LR fuzzy number uLR = (1.0, 0.6, 6.0, 1.3; 3.0, 1.4, 3.0, 1.7) and its LU approximation uLR = (1.0, 0.6, 6.0, 1.3; 3.0, 1.4, 3.0, 1.7) obtained by the mixed spline with N = 1. The shape functions L(x) and R(x), x ∈ [1, 6] are reported with inverted axes (y, x), horizontal x, to meet the Lower/Upper + inverses u−
, u
,
∈ [0, 1]. The (LU)–(LR) fuzzy relationship above can be used as an intermediate step for the LR-fuzzy calculus. For simplicity, we describe only the multiplication of two positive LR-fuzzy numbers u and v uLR = (u0,L , d0,L , u0,R , d0,R ; u1,L , d1,L , u1,R , d1,R ), u0,L > 0, vLR = (v0,L , e0,L , v0,R , e0,R ; v1,L , e1,L , v1,R , e1,R ), v0,L > 0. Their LU-representations are − + + − − + + uLU = (u− 0 , u0 , u0 , u0 ; u1 , u1 , u1 , u1 ), vLU = (v0− , v0− , v0+ , v0+ ; v1− , v1− , v1+ , v1+ ) ± ± ± with u± i , vi , ui and vi (i = 0, 1) calculated according to (23). For the (approximated) LU-fuzzy multiplication w = uv we obtain − − − − − + + + + + + wLU = (u− 0 v0 , u0 v0 + u0 v0 , u0 v0 , u0 v0 + u0 v0 ; − − − − − − + + + + + + u1 v1 ,u1 v1 + u1 v1 ,u1 v1 ,u1 v1 + u1 v1 )

and, returning to the LR-fuzzy form of w , we get the LR multiplication  − − − + + + u− u+ − + + 1 v 1 − u 0 v0 1 v1 − u 0 v 0 wLR = u− , u v , 0 v0 , 0 0 − − − + + +, u− u+ 1 v1 + u1 v1 1 v1 + u1 v1  − − − + + + u− u+ − − + + 1 v 1 − u 0 v0 1 v1 − u 0 v0 u1 v1 , − − − , u1 v1 , + + + . u− u+ 0 v0 + u0 v0 0 v0 + u0 v0 Fig. 4 depicts two nonnegative LR-fuzzy numbers and their product w = uv obtained as above.

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L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

5. LU-fuzzy calculus In this section we will consider the application of the LU-fuzzy representation to different calculations that appear in many areas of the fuzzy calculus, including defuzzification (ranking functions), fuzzy differentiation and integration, fuzzy linear equations, fuzzy linear least squares regression, fuzzy differential equations and Zadeh extension of functions. In all the calculations where it is explicitly required, we will use the mixed model in one of the two forms (8) or (10) with N = 1. The general case with N > 1 can be obtained easily. 5.1. Ranking functions The ordering of fuzzy numbers can be approached in many ways (see [34] or [21] for recent literature and results); in most cases, the fuzzy numbers u are transformed into real numbers. By the use of the LU-fuzzy representation, the ranking functions can be computed, either numerically or, if possible, analytically, in terms of the parameters u± i and that define u. u± i The first example is the so-called level set average [17], defined by

1 1 − u∗DP = (u + u+
) d
; 2 0
the calculations give u∗DP

1 1 − + = (u− + u+ 0 + u1 + u1 ) + 4 0 4



− + u+ u− 0 − u1 0 − u1 + a− a+



1 + 2



 − + u− u+ 1 − u0 1 − u0 + , a − (a − + 1) a + (a + + 1)

where a− = 1 +

− u− 0 + u1 − u− 1 − u0

and a+ = 1 +

+ u+ 0 + u1 + u+ 1 − u0

Related to the calculation of the above integral is the nearest interval approximation of a fuzzy number, described by Grzegorzewski in [29], expressed equivalently either in terms of the lower–upper functions or in terms of the membership function

 1  

1

u−

u+ 1 0 − + − + C(u) = u
d
, u
d
= u1 − (x) dx, u1 + (x) dx ; 0

u− 0

0

u+ 1

we obtain

   − − u− u− 1 − − 0 − u1 1 − u0 C(u) = + u0 + u1 + , 2 a− a − (a − + 1)  

+ + u+ u+ 1 + + 0 − u1 1 − u0 u0 + u1 + + + + . 2 a+ a (a + 1)

A second example is the interval valued possibilistic mean M(u) and the level-weighted average u∗GW (see Goetschel and Voxman [25] and Carlsson and Fullér [3]) given by M(u) = [M − (u), M + (u)] where

1

− − +
u
d
, M (u) = 2 M (u) = 2 0

0

1


u+
d


L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

2437

and u∗GW =



1 0

+
(u−
+ u
) d
=

M − (u) + M + (u) ; 2

the calculations give −

M (u) =

u− 0

  − 2u− 2u− u− u− 7 − − 0 0 + u1 1 0 + u1 − u0 − − , + − + a 10 a− a − (a − + 2) a − (a − + 1)(a − + 2)

  + + + + u 2u+ u u 2u+ 7 + + 0 0 1 1 0 + u M + (u) = u+ + − u − − 0 + 1 0 a+ 10 a+ a + (a + + 2) a + (a + + 1)(a + + 2) so that 1 1 u− 1 u+ + 0 0 u∗GW = (u− + 0 + u0 ) + − + 2 2 a 2 a     − + u− u+ 7 7 − − + + 0 + u1 0 + u1 + + u1 − u0 − u1 − u0 − a− a+ 20 20 +

u− u+ u− u+ 1 1 0 0 + − − . a − (a − + 2) a + (a + + 2) a − (a − + 1)(a − + 2) a + (a + + 1)(a + + 2)

Other functions can be calculated numerically; examples are the covariance and variance of fuzzy numbers u, v (see [3,24]); their analytical expressions are very long and a numerical integration is probably preferred. The same can be done for the Yager and Filev weighted centroid (see [59]). 5.2. LU-fuzzy random numbers The LU-fuzzy representation can be useful to describe fuzzy random variables (see [9,10,49] or [57] and the references therein). In a way similar to the well known LR-fuzzy random variables, we start with a given probability space + − + (, , P ) and consider the parameters u− i , ui , ui , ui :  → R as random variables satisfying (15), (16) and (17) for i = 0, 1, . . . , N; it follows that each − + + u() = (u− i (), ui (), ui (), ui ())i=0,1,...,N ,

∈

defines a valid fuzzy number and u :  →  FN is a LU-fuzzy random number over (, , P ); a similar situation − − + + arises in the general nondifferentiable model. In the case of model (10), the parameters (n− i , i , ui (); ni , i , + − + − + ui ())i=0,1,...,N are required; if ni = ni = n and i = i =  have fixed (nonrandom) values for all i, then only + u− i () and ui () are needed as random variables. + As the support function of each [u−
(), u
()] is given over {−1, 1} by s[u−
(),u+
()] (−1) = −u−
(),

s[u−
(),u+
()] (1) = u+
()

then the same results of [10] or [57] can be applied to our case. In particular, the expectation and the conditional expectation for LU-fuzzy random variables are obtained, respectively, by − + + Eu = (Eu− i , Eui , Eui , Eui )i=0,1,...,N

and, for each conditioning sub-algebra Υ ⊂  − + + E(u|Υ ) = (E(u− i |Υ ), E(ui |Υ ), E(ui |Υ ), E(ui |Υ ))i=0,1,...,N .

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L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

5.3. Integration and differentiation of fuzzy-valued functions Integrals and derivatives of fuzzy-valued functions have been established, among others, by Dubois and Prade [16], Kaleva [35,36] and Puri and Ralescu [49]; see also [38] for some recent results. We consider here a function u : [a, b] −→  FN where u(t) = (u− (t), u+ (t)) for t ∈ [a, b] is an LU-fuzzy number of the form − + + u(t) = (u− i (t), ui (t), ui (t), ui (t))i=0,1,...,N .

The integral of u(t) with respect to t ∈ [a, b] is given by  b  b 

b − + u(t) dt = u
(t) dt, u
(t) dt , v
:=


a

a


∈ [0, 1].

a

In the LU-fuzzy representation, the integral can be written as v = (vi− , vi− , vi+ , vi+ )i=0,1,...,N , where vi±



b

= a

u± i (t) dt

and

vi±



b

= a

u± i (t) dt, i = 0, 1, . . . , N.

± More generally, if u± i and ui are -measurable functions over A then 







− − + + u d = ui d , ui d , ui d , ui d , A

A

A

A

A

.

i=0,1,...,N

The (Hukuhara) derivative of the fuzzy-valued function u(t) at a point  t is obtained by the derivatives of the lower and upper branches of the
-cuts     d −  d +   (u (t))
= u (t) u (t) , dt
t=t dt
t=t provided that the intervals define a correct fuzzy number for each t, that is to say, the following conditions hold: d each u− (t) is nondecreasing w.r.t.
, dt
d each u+ (t) is nonincreasing w.r.t.
, dt
d − d (t), ∀
∈ [0, 1]. (24) u
(t)
u+ dt dt
Using the LU-fuzzy representation, we obtain (the  means derivative w.r.t. t) −  +  +   u ( t) = (u− i (t), ui (t), ui (t), ui (t))i=0,1,...,N

and the conditions for a valid fuzzy derivative are, for i = 0, 1, . . . , N −  −   u− 0 (t)
u1 (t)
· · ·
uN (t) +  +  
u+ N (t)
uN−1 (t)
· · ·
u0 (t),

 u− i (t)  0, +  ui (t)
0.

(25)

The following examples are from [23]. Consider first u
(t) = [
t 2 , (2 −
2 )t] so that the exact Hukuhara derivative is u
(t) = [2
t, 2 −
2 ] and u (t) is a valid fuzzy number for 0
t
21 . To test our model, we select N = 1 so

L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

2439

Fig. 5. LU-fuzzy Hukuhara derivative (first example).

that the (t-dependent) 8-vector representation is u(t) = (0, t 2 , 2t, 0; t 2 , t 2 , t, −2t); the error is of the order of 2.0 × 10−16 . Fig. 5 describes the LU-fuzzy Hukuhara derivative u (t) = (0, 2t, 2, 0; 2t, 2t, 1, −2) of the fuzzy function u(t) = (0, t 2 , 2t, 0; t 2 , t 2 , t, −2t). For different values of t ∈ [0, 21 ], the membership functions of u (t) “move” from left to right (the vertical axis contains the membership grades between 0 and 1); so, for example, the last curve bounding the surface on the right is the fuzzy number u ( 21 ) = (0, 1, 2, 0; 1, 1, 1, −2). Considering a second example u
(t) = [e
t , 4t −
2 + 1], the exact Hukuhara derivative is u
(t) = [
e
t , 4t ln(4)] and u (t) is a valid fuzzy number for t 0. With N = 1, the representation is u(t) = (1, t, 4t + 1, 0; et , tet , 4t , −2) and the derivative has the representation u (t) = (0, 1, 4t ln(4), 0; et , (t + 1)et , 4t ln(4), 0). It is represented in Fig. 6. The error is null for the left branches and is less than 0.001 for the right branches. Fig. 6 describes the LU-fuzzy Hukuhara derivative u (t). For different values of t ∈ [0, 1], the membership functions of u (t) “move” from left to right (the vertical axis contains the membership grades between 0 and 1); so, for example, the last curve bounding the surface on the right is the fuzzy interval u (1) = (0, 1, 4 ln(4), 0; e, 2e, 4 ln(4), 0). Note that the upper branch of u (t) is constant for all t and produces a “vertical” portion of the membership function. 5.4. Computation of fuzzy-valued functions In [30] we have analyzed the advantages of the LU-representation in the computation of fuzzy expressions by the direct application of the interval arithmetic operations (INT ) or by the constrained fuzzy arithmetic (CFA) method (see [40,41]). In this context, fuzzy expressions like z = (u2 + v)(v + w 2 ), for given fuzzy numbers u, v and w, can be obtained either by the application of the standard interval arithmetic, where each occurrence of a given fuzzy quantity is handled independently, or by the CFA method (we denote by zINT and zCFA the corresponding results); also a mixed approach (see [47]) is frequently used, e.g. zMIX = ((u2 )CFA + v)(v + (w 2 )CFA ), where only the squares are computed via CFA (or, equivalently, via the unary square operator) and the other operations are executed by interval arithmetic; it is well known that, in general, (zCFA )
⊆ (zMIX )
⊆ (zINT )
. More generally, we can use the LU-representation in the calculation of fuzzy extensions of functions, in the Zadeh general extension principle setting (EP). Let v = f (u1 , u2 , . . . , un ) denote the fuzzy extension of a continuous function

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L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

Fig. 6. LU-fuzzy Hukuhara derivative (second example).

f in n variables; it is well known that the fuzzy extension of f to normal upper semicontinuous fuzzy intervals (with compact support) has the level-cutting commutative property (see [14]), i.e. the
-cuts [v
− , v
+ ] of v are the images of the
-cuts of (u1 , u2 , . . . , un ) and are then obtained by solving the following box-constrained optimization problems for
∈ [0, 1]:  − + v
= min{f (x1 , x2 , . . . , xn )|xk ∈ [u− k,
, uk,
], k = 1, 2, . . . , n}, (EP) : (26) + v
+ = max{f (x1 , x2 , . . . , xn )|xk ∈ [u− k,
, uk,
], k = 1, 2, . . . , n}. In this case the
-cuts of z = (u2 + v)(v + w 2 ) above are EP + (zEP )
= [(zEP )−
, (z )
]

where

2 2 (zEP )−
= min{(xu + xv )(xv + xw )|xu ∈ u
; xv ∈ v
; xw ∈ w
}, 2 2 (zEP )+
= max{(xu + xv )(xv + xw )|xu ∈ u
; xv ∈ v
; xw ∈ w
}.

Except for simple elementary cases for which the optimization problems above can be solved analytically, the direct application of (EP) is difficult and computationally expensive as, for each
∈ [0, 1], the global solutions of two nonlinear programming problems are required. Different approaches have been proposed to solve the problem: among others, the vertex method and its variants (see [5,12,48]), the fuzzy weighted average method (see [13]), the general transformation method (see [31,32]), the interval arithmetic optimization with sparse grids approximation of the objective function (see [39]). At least in the differentiable case, the advantages of the LU-representation appear to be quite interesting, based on the fact that a small number of
points is in general sufficient to obtain good approximations (this is the essential gain in using the slopes to model fuzzy numbers), so reducing the number of constrained min and max problems to be solved directly.

L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

2441

In the following subsections, we give the details of the fuzzy extension of general (piecewise) differentiable functions by the LU-representation. In all the computations we will adopt the EP method, but also if other approaches are adopted, the representation still remains valid. We consider first the single variable case. 5.4.1. Univariate functions To see how the LU-representation works, we consider first a single variable differentiable function f : R → R; its + (EP)-extension v = f (u) to a fuzzy argument u
= [u−
, u
] has
-cuts v
= [min{f (x)|x ∈ u
}, max{f (x)|x ∈ u
}].

(27)

+ + − If f is monotonic increasing we obtain v
= [f (u−
), f (u
)] while, if f is monotonic decreasing, v
= [f (u
), f (u
)]; for this simple situations, we introduce a notation similar to the one used for multiplication and division: define q
− , q
+ ∈ {−, +} by  + − − if min{f (u−
), f (u
)} = f (u
), − q
= + + + if min{f (u−
), f (u
)} = f (u
),  + − − if max{f (u−
), f (u
)} = f (u
), + q
= + + + if max{f (u−
), f (u
)} = f (u
). q−

q+



+ ± ± We have f (u)−
= f (u
) and f (u)
= f (u
); we simplify the notation q
i ≡ qi in the points of the
− − + + − decomposition. If u = (ui , ui , ui , ui )i=0,1,...,N then the LU-representation v = (vi , vi− , vi+ , vi+ )i=0,1,...,N of its image v = f (u) is

q−

q−

q−

q+

q+

q+

v = (f (ui i ), f  (ui i )ui i , f (ui i ), f  (ui i )ui i )i=0,1,...N . As an example, the monotonic exponential function f (x) = exp(x) has fuzzy extension (qi− = −, qi+ = +) − − + + + v = exp(u) = (exp(u− i ), exp(ui )ui , exp(ui ), exp(ui )ui )i=0,1,...N .

In the nonmonotonic (differentiable) case, we have to solve the optimization problems in (27) for each
=
i , i = 0, 1, . . . , N, i.e.  − + vi = min{f (x)|x ∈ [u− i , ui ]}, (EPi ) : + vi+ = max{f (x)|x ∈ [u− i , ui ]}. + The min and the max of (EPi ) can occur either at a point which is internal to the corresponding interval [u− i , ui ] or − + xi the points where the min and the max in (EPi ) take coincident with one of the extremal values; denote by  xi and  − + place and introduce the indices qi and qi by ⎧ xi− := u− ⎪ i , ⎨ − if min is taken at  − − qi = + if min is taken at  xi := u+ i , ⎪ ⎩ + ∗ if min is taken at internal point  xi− ∈]u− i , ui [, ⎧ xi+ := u− ⎪ i , ⎨ − if max is taken at  + + qi = + if max is taken at  xi := u+ i , ⎪ ⎩ + ∗ if max is taken at internal point  xi+ ∈]u− i , ui [.

Then, for each
i , we obtain vi− = f ( xi− ) and vi+ = f ( xi+ ) and vi− , vi+ are computed by   q− q+ f  ( f  ( xi− )ui i if qi− = − or + xi+ )ui i if qi+ = − or + − + vi = and vi = . 0 if qi− = ∗ 0 if qi+ = ∗

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L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

Fig. 7. Membership function of the LU-fuzzy number u = (−1, 1, 1, −1; 0.5, 1, 0.5, 1).

Fig. 8. Membership function of the fuzzy extension of the nonmonotonic function v = u4 .

As a nonmonotonic example, consider the power function f (x) = x n with positive (odd or even) integer exponent n; − + + n if u = (u− i , ui , ui , ui )i=0,1,...,N , the LU representation of u is − n−1 − + n−1 + n n ui , (u+ ui )i=0,1,...N un = ((u− i ) , n(ui ) i ) , n(ui )

if n is odd, or, if n is even, ⎧ − n n−1 u− , (u+ )n , n(u+ )n−1 u+ ) ((ui ) , n(u− ⎪ i ) i i i i ⎪ ⎪ ⎪ ⎨ ((u+ )n , n(u+ )n−1 u+ , (u− )n , n(u− )n−1 u− ) i i i i i i un = + n + n−1 + ⎪ (0, 0, (u ) , n(u ) u ) ⎪ i i i ⎪ ⎪ ⎩ − n − n−1 − (0, 0, (ui ) , n(ui ) ui )

if u− i 0, if u+ i
0,

+ − + if u− i
0, ui 0 and |ui |
|ui |,

+ − + if u− i
0, ui 0 and |ui | > |ui |.

Figs. 7 and 8 illustrate the LU-fuzzy number u = (−1, 1, 1, −1; 0.5, 1, 0.5, 1) and its image in the fuzzy extension u4 . The exact membership function is evaluated at 101 equidistant
-points; the average absolute error of the mixed-spline model with N = 5 subintervals is 0.19% and it improves to 0.03% with N = 11. Fig. 7 describes the membership function of the LU-fuzzy number u = (−1, 1, 1, −1; 0.5, 1, 0.5, 1) obtained by the mixed spline model. u is a fuzzy number with compact support [−1, 1]. The
-cuts of u contain elements which are all positive only for
∈ [
, 1] with
 0.65.

L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

2443

Fig. 8 describes the membership function of the fuzzy extension of the nonmonotonic function v = u4 applied to u = (−1, 1, 1, −1; 0.5, 1, 0.5, 1). The extension principle (EP) is adopted; the images of the
-cuts of u having a negative part are of the form [0, s 4 ], producing a constant portion in the lower branch v
− for
∈ [0,
] with
 0.65. The graphs report the LU representation obtained with N = 5 (6 points) and the exact membership function of v. The corresponding average error is 0.19%. 5.4.2. Multivariate functions Consider now the extension of a multivariate differentiable function f : Rn → R to a vector of n fuzzy numbers u = (u1 , u2 , . . . , un ) with kth component − + + uk = (u− k,i , uk,i , uk,i , uk,i )i=0,1,...,N

for k = 1, 2, . . . , n.

Let v = f (u1 , u2 , . . . , un ) and v = (vi− , vi− , vi+ , vi+ )i=0,1,...,N be its LU-representation; the
-cuts of v are obtained by solving the box-constrained optimization problems (26). For each
=
i , i = 0, 1, . . . , N the min and the max (26) can occur either at a point xk which is internal to the + x − = ( x1− ,  x2− , . . . ,  xn− ) corresponding interval [u− k,i , uk,i ] or it is coincident with one of the extremal values; denote by  and  x + = ( x1+ ,  x2+ , . . . ,  xn+ ) the points where the min and the max take place (or approximations to them obtained by − − − + + + one of the mentioned methods) and introduce the indices (qi,1 , qi,2 , . . . , qi,n ) and (qi,1 , qi,2 , . . . , qi,n ) by ⎧ − − if min, for
=
i , is taken at  xk,i = u− ⎪ k,i , ⎨ − − xk,i = u+ qi,k = + if min, for
=
i , is taken at  (28) k,i , ⎪ ⎩ − − + ∗ if min, for
=
i , is taken at  xk,i ∈]uk,i , uk,i [, ⎧ + − if max, for
=
i , is taken at  xk,i := u− ⎪ k,i , ⎨ + + xk,i := u+ qi,k = + if max, for
=
i , is taken at  (29) k,i , ⎪ ⎩ + − + ∗ if max, for
=
i , is taken at  xk,i ∈]uk,i , uk,i [. Then we obtain − − − x1,i , x2,i , . . . , xn,i ) vi− = f (

+ + + and vi+ = f ( x1,i , x2,i , . . . , xn,i )

and the slopes vi− , vi+ are computed according to (as f is differentiable) − − − n − jf ( x1,i , x2,i , . . . , xn,i ) qi,k vi− = uk,i , jxk k=1 −

=∗ qi,k

vi+ =

+ + + n jf ( x1,i , x2,i , . . . , xn,i ) k=1 + qi,k

=∗

jxk

q+

uk,ii,k .

(30)

As a first example, we consider the following (from [40]) f : R2 → R defined by f (x1 , x2 ) = x1 x2 /x1 + x2 to be extended to v = f (u1 , u2 ) with two triangular fuzzy numbers u1 = 0, 1, 2 , u2 = 1, 2, 3. The
-cuts are u1,
= [
, 2 −
] , u2,
= [1 +
, 3 −
] so that (u1 u2 )
= [
(1 +
),
2 − 5
+ 6], (u1 + u2 )
= [1 + 2
, 5 − 2
] and the extensions v = f (u1 , u2 ) are  
(1 +
)
2 − 5
+ 6 v
CFA = , , 1 + 2
5 − 2
 
(1 +
)
2 − 5
+ 6 v
INT = , . 5 − 2
1 + 2
The LU representations of u1 and u2 are obtained exactly with N = 1 and u1 = (0, 1, 2, −1; 1, 1, 1, −1), u2 = (1, 1, 3, −1; 2, 1, 2, −1) so that u1 + u2 = (1, 2, 5, −2; 3, 2, 3, −2) and u1 u2 = (0, 1, 6, −5; 2, 3, 2, −3). The 2point LU-fuzzy approximation of v INT is (0, 0.2, 6, −17; 0.67, 1.44, 0.67, −1.44) with an average error of 2.01% in

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L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

the mixed-spline model (calculated in 101 equidistant points
∈ [0, 1]). To obtain the CFA solution, observe that the gradient vector of f is (x22 /(x1 + x2 )2 , x12 /(x1 + x2 )2 ) (nonnegative) so that v EP = v CFA ; for
= 0 v0− = min{f (0, 1), f (0, 3), f (2, 1), f (2, 3)} = 0, v0+ = max{f (0, 1), f (0, 3), f (2, 1), f (2, 3)} =

6 5

while, for
= 1, v1− =

and v1+ = 23 .

2 3

To obtain the 2-point LU representation of v CFA we have to compute the slopes by (30), giving v0− = 1,

v0+ = − 13 25

and v1− = 59 ,

v1+ = − 59

so that the approximated v EP = v CFA is (0, 1, 1.2, −0.52; 0.67, 0.56, 0.67, −0.56), with an average error of 1.23% in the mixed spline model. We now consider the fuzzy extension of the following nonmonotonic differentiable functions (from [31,39]): f : (x1 , x2 ) −→ x2 cos( x1 ),

x1 ∈ [0, 5], x2 ∈ [1, 5]

and g : (x1 , x2 ) −→

1 , 0.2 + (x1 − 2)4 + (x2 − 2)2

x1 ∈ [0, 5], x2 ∈ [1, 5].

As in [39] we apply f and g to the symmetric triangular fuzzy numbers u1 = 0, 2.5, 5 and u2 = 1, 3, 5, having
-cuts u1,
= [2.5
, 5 − 2.5
] and u2,
= [1 + 2
, 5 − 2
]. We use the LU representation with N = 5 subintervals; the data are represented in the following table: i 0 1 2 3 4 5

− +
i u− 1,i u1,i u1,i 0.0 0.0 2.5 5.0 0.2 0.5 2.5 4.5 0.4 1.0 2.5 4.0 0.6 1.5 2.5 3.5 0.8 2.0 2.5 3.0 1.0 2.5 2.5 2.5

− − + u+ 1,i u2,i u2,i u2,i −2.5 1.0 2.0 5.0 −2.5 1.4 2.0 4.6 −2.5 1.8 2.0 4.2 −2.5 2.2 2.0 3.8 −2.5 2.6 2.0 3.4 −2.5 3.0 2.0 3.0

u+ 2,i −2.0 −2.0 −2.0 −2.0 −2.0 −2.0

and the details of the calculations of the extension of the first function f , according to (26), (28) and (29) are the following: i 0 1 2 3 4 5


i 0.0 0.2 0.4 0.6 0.8 1.0

− − − − + + + +     x1,i x2,i x1,i x2,i q1,i q2,i q1,i q2,i 1.0 5.0 ∗ + 0.0 5.0 ∗ + 1.0 4.6 ∗ + 2.0 4.6 ∗ + 1.0 4.2 ∗ + 4.0 4.2 ∗ + 3.0 3.8 ∗ + 2.0 3.8 ∗ + 3.0 3.4 ∗ + 2.0 3.4 ∗ + 2.5 3.0 + + 2.5 3.0 + +

The application of (30) produces the final result v = f (u1 , u2 ) i 0 1 2 3 4 5


i 0.0 0.2 0.4 0.6 0.8 1.0

vi− vi− −5.0 2.0 −4.6 2.0 −4.2 2.0 −3.8 2.0 −3.4 2.0 0.0 23.562

vi+ vi+ 5.0 −2.0 4.6 −2.0 4.2 −2.0 3.8 −2.0 3.4 −2.0 0.0 −23.562

L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

2445

Fig. 9. Fuzzy extension of f (u1 , u2 ) = u2 cos( u1 ) with triangular u1 = 1, 2.5, 5, u2 = 1, 3, 5. The extension is calculated by the application of the (EP) method to the LU representation of the arguments with a
-decomposition into N = 5 subintervals. Note that the function is not monotonic with respect to the second parameter.

Fig. 10. Fuzzy extension of v = g(u1 , u2 ) = 1/0.2 + (u1 − 2)4 + (u2 − 2)2 with triangular u1 = 1, 2.5, 5, u2 = 1, 3, 5. The extension is calculated by the application of the (EP) method to the LU representation of the arguments with a
-decomposition into N = 5 subintervals. Note that the function is not monotonic and has a local internal maximum point producing a partially constant upper branch (and a vertical portion on the right of the membership function). For high values of
, greater than about 0.5, the local maximum point is not internal to the
-cuts and the lower and upper branches v
− and v
+ are both strictly monotonic.

Similar calculations are performed to obtain the LU representation (N = 5) of the second function v = g(u1 , u2 ) i 0 1 2 3 4 5


i 0.0 0.2 0.4 0.6 0.8 1.0

vi− 0.011 0.022 0.048 0.118 0.316 0.792

vi− vi+ vi+ 0.035 5.0 0.0 0.079 5.0 0.0 0.201 5.0 0.0 0.566 4.167 −13.889 1.562 1.786 −7.653 3.294 0.792 −3.294

Figs. 9 and 10 illustrate the results obtained by the mixed spline model. 5.5. Fuzzy differential equations The fuzzy differential (and integral) equations (FDE) have been extensively described and analyzed in the fuzzy literature. There also exist different approaches and solution concepts. It is out of the scope of this paper to enter details

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L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

about the theory of FDE (some recent results and references can be found for example in [1,2,23,26,45,46]); we are interested here to show that the LU-representation can be useful at least at a computational level, as we can approximate the FDE by a finite set of ordinary (nonfuzzy) differential equations (ODE) with a possibly controllable precision, by increasing eventually the number of points of the
-cut decomposition. For simplicity, we consider the first-order unidimensional initial value fuzzy differential equation (Cauchy problem) d u(t) = f (t, u(t)), dt u(t0 ) = u(0) , (31) where f : [t0 , T ] × F −→ F is the fuzzy extension of a real-valued differentiable function y = f (t, x), t ∈ [t0 , T ], x ∈ R and u(0) ∈ F is a given fuzzy initial condition. A solution is a fuzzy function u : [t0 , T ] −→ F satisfying t (d/dt)u(t) = f (t, u(t)), ∀t ∈ [t0 , T ] or, in integral form, u(t) = u(0) + t0 f (s, u(s)) ds. (0)−

Let u(0) = (ui

(0)−

, ui

(0)+

, ui

(0)+

, ui

)i=0,1,...,N be the LU-fuzzy initial condition and let u(t) have the form

− + + u(t) = (u− i (t), ui (t), ui (t), ui (t))i=0,1,...,N .

We can calculate u(t) by solving the following 2(N + 1) ODEs, for i = 0, 1, . . . , N: d − u (t) = (f (t, u(t)))− i , dt i d + u (t) = (f (t, u(t)))+ i , dt i (0)± u± i (t0 ) = ui

(32)

for the levels, where the extensions (f (t, u))± i are obtained by the EP method − + (f (t, u))− i = min{f (t, x)|x ∈ [ui (t), ui (t)]},

− + (f (t, u))+ i = max{f (t, x)|x ∈ [ui (t), ui (t)]}.

(33)

To determine the corresponding slopes (if model (9)–(10) is not preferred) we add to (32) the following 2(N + 1) ODEs: d u− (t) = (f (t, u(t)))− i , dt i d u+ (t) = (f (t, u(t)))+ i , dt i (0)± , (34) u± i (t0 ) = ui where the (f (t, u(t)))± i are computed according to a rule similar to (28), (29) and (30), with the appropriate values identified in (33). The multidimensional case can be approached in a similar way. If we adopt a model in the form (9)–(10), the solution of the ODEs involving the slopes can be avoided, but loosing precision in the approximations. An advantage of the LU-representation is also related to the possibility of verifying the time-domain t ∈ [t0 , T ] where the FDE (31) admits a solution, by testing the validity of (25) for each t of interest. To give some first examples, we consider cases where the fuzzy solution can be obtained analytically so as to compare the computations with the exact solutions. 5.5.1. Examples with analytical solution Consider the simple linear homogeneous FDE with a triangular fuzzy initial condition (from [46]) u (t) = u(t), t ∈ [0, T ], u
(0) = [0.75 + 0.25
, 1.125 − 0.125
]. The exact solution is given by u
(t) = et [0.75 + 0.25
, 1.125 − 0.125
]; the LU-fuzzy representation with N = 1 is − + + exact and gives u(0) = (0.75, 0.25, 1.125, −0.125; 1, 0.25, 1, −0.125). If u(t) = (u− i (t), ui (t), ui (t), ui (t))i=0,1

L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

and we model u(t) with n = 1, =

1 2 in (9)–(10), the four (32) independent equations are sufficient − − u− 0 (t) = u0 (t) with initial u0 (0) = 0.75, + + u+ 0 (t) = u0 (t) with initial u0 (0) = 1.125, − − u1 (t) = u1 (t) with initial u− 1 (0) = 1, + + u1 (t) = u1 (t) with initial u+ 1 (0) = 1 − + + t t obtaining u0 (t) = 0.75e , u0 (t) = 1.125et , u− 1 (t) = u1 (t) = e . The LU-fuzzy solution is then

2447

to be solved, i.e.

u(t) = (0.75et , 0.25et , 1.125et , −0.25et ; et , 0.25et , et , −0.25et ) which is exact. Note that if we solve the ODEs (34) for the slopes we obtain the same solution. Consider now the linear FDE (from [26]) u (t) + u(t) = v(t), u
(0) = [
− 1, 1 −
], where v(t) ∈ F is a given fuzzy function with
-cuts v
= e−t [
− 1, 1 −
]. The exact solution is the triangular fuzzy function u
(t) = e−t (1 + t)[
− 1, 1 −
]. In the LU-fuzzy context, with N = 1, we have the following representations: v(t) = e−t (−1, 1, 1, −1; 0, 1, 0, −1), − + + u(t) = (u− i (t), ui (t), ui (t), ui (t))i=0,1 . Also in this case, we can model u(t) with n = 1,  =

The

1 2

in (9)–(10), and solve only the four (32) independent ODEs

− −t u− with initial u− 0 (t) + u0 (t) = −e 0 (0) = −1, + + + −t with initial u0 (0) = 1, u0 (t) + u0 (t) = e − u− (t) + u (t) = 0 with initial u− 1 1 1 (0) = 0, + + u1 (t) + u1 (t) = 0 with initial u+ 1 (0) = 0. − −t −t solutions are u0 (t) = −(1 + t)e , u+ 0 (t) = (1 + t)e ,

+ u− 1 (t) = u1 (t) = 0 so that u(t) is given by (it is exact)

u(t) = (−(1 + t)e−t , (1 + t)e−t , (1 + t)e−t , −(1 + t)e−t ; 0, (1 + t)e−t , 0, −(1 + t)e−t ). 5.5.2. Examples with numerical solution ± In the next example, we solve an FDE exactly for the levels u± i (t) and numerically for the slopes ui (t). Consider the FDE with crisp initial condition (from [1]) u (t) = a u2 (t) + b, u
(0) = [0, 0],

t ∈ [0, 21 ],

where a and b are constant positive triangular fuzzy numbers with a
= [
, 2 −
] and b
= [1 +
, 3 −
]. Observe first that all the quantities are nonnegative, the
-cuts of the fuzzy function f (u; a, b) = a u2 + b are obtained by 2 + 2 the standard arithmetic: f
(u; a, b) = [
(u−
) +
+ 1, (2 −
)(u
) + 3 −
]; the partial derivatives of f (u; a, b) are ju f (u; a, b) = 2au, ja f (u; a, b) = u2 , jb f (u; a, b) = 1. The LU-fuzzy representation with N = 1 gives − + + u(0) = (0, 0, 0, 0; 0, 0, 0, 0) and u(t) = (u− i (t), ui (t), ui (t), ui (t))i=0,1 . The four (32) ODEs are − u− 0 (t) = f0 (u) = 1

with initial u− 0 (0) = 0,

+ + 2 u+ 0 (t) = f0 (u) = 2(u0 ) + 3 − − 2 u− 1 (t) = f1 (u) = (u1 ) + 2

+ + 2 u+ 1 (t) = f1 (u) = (u1 ) + 2

with initial u+ 0 (0) = 0,

with initial u− 1 (0) = 0, with initial u+ 1 (0) = 0

and the four (34) ODEs for the slopes, according to (30), are − 2 u− 0 (t) = (u0 (t)) + 1

with initial u− 0 (0) = 0,

+ + + 2 u+ 0 (t) = 4u0 (t)u0 (t) − (u0 (t)) − 1

with initial u+ 0 (0) = 0,

+ + + 2 u+ 1 (t) = 2u1 (t)u1 (t) − (u1 (t)) − 1

with initial u+ 1 (0) = 0.

− − − 2 u− 1 (t) = 2u1 (t)u1 (t) + (u1 (t)) + 1

with initial u− 1 (0) = 0,

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L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

Fig. 11. Fuzzy numerical solution u(t) of u (t) = au2 (t) + b for different values of t = 0.1, 0.2, 0.3, 0.4, 0.5 (for increasing t the curves “move” from bottom to top). The initial condition is nonfuzzy u(0) = 0 and the parameters a and b are the triangular fuzzy numbers a = 0, 1, 2 and b = 1, 2, 3; the LU representations of u(t) are calculated with the mixed spline model and N = 1. The membership values are reported in the horizontal axis.

+ Note that u− 1 (t) = u1 (t) and  √ + 3 u− (t) = t, u (t) = 0 0 2 tan(t 6), √ √ + u− 1 (t) = 2 tan(t 2) = u1 (t).

The computed solution at times t = 0.1, 0.2, 0.3, 0.4, 0.5 is reported in Fig. 11. In the last example we calculate numerical solutions both for levels and slopes. Consider the FDE with triangular initial condition (from [46]) u (t) = e−u

2 (t)

,

t ∈ [0, 21 ],

u
(0) = [0.75 + 0.25
, 1.5 − 0.5
]. The fuzzy function f (u) = e−u has
-cuts (using EP) ⎧ ⎪ e−(u−
)2 if u+ ⎪

0, ⎨ + )2 − −(u f
(u) = e
if u−
0, ⎪ ⎪ − )2 + )2 ⎩ −(u −(u min{e
, e
} otherwise, ⎧ −(u+ )2
if u+ ⎪

0, ⎨e − 2 + f
(u) = e−(u
) if u−
0, ⎪ ⎩ 1 otherwise. 2

The LU-fuzzy representation with N = 1 gives − + +
i u− i (0) dui (0) ui (0) dui (0) u(0) = 0 0.75 0.25 1.5 −0.5

1

1

0.25

1

−0.5

L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

2449

Fig. 12. Fuzzy numerical solution u(t) of u (t) = exp(−u2 (t)) with triangular fuzzy initial condition u(0) = 0.75, 1, 1.5, for values of t = 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5 (for increasing t the curves “move” from bottom to top). The LU representations of u(t) are calculated with the mixed spline model and N = 1. The membership values are reported in the horizontal axis.

− + + and u(t) = (u− i (t), ui (t), ui (t), ui (t))i=0,1 , so we have four (32) ODEs for the levels and four (34) ODEs for the slopes. The numerical solution is obtained by the use of a standard ODE solver and is illustrated in Fig. 12.

5.6. Fuzzy linear equations and least squares regression As a final application of LU-fuzzy representation, we consider a system of simultaneous fuzzy linear equations with crisp coefficients and fuzzy right-hand side, as discussed in detail by Friedman et al. (see [22]). We also consider briefly the applicability of the LU-representation in the least squares estimation of a linear fuzzy model (see [4,10,20,58,37] where nontriangular fuzzy observations are considered). In this section, a vector of LU-fuzzy numbers (over the same
-discretization), with n components, u = (u1 , u2 , . . . , un ) is represented as − + + u = (u− k,i , uk,i , uk,i , uk,i )i=0,1,...,N ,

k = 1, 2, . . . , n.

An n-dimensional fuzzy simultaneous linear system Ax = b whose coefficients A = [akj ] are crisp and x and b are LU-fuzzy, can be written as ⎧ n − ⎪ (akj xj )− ⎪ ⎨ i = bk,i j =1 , k = 1, 2, . . . , n, for the levels, and n  ⎪ + ⎪ (akj xj )+ = b ⎩ i k,i j =1

⎧ n − ⎪ (akj xj )− ⎪ ⎨ i = bk,i j =1

, n  ⎪ + ⎪ (akj xj )+ = b ⎩ i k,i j =1

k = 1, 2, . . . , n, for the slopes,

(35)

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L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

where

 (akj xj )± i

=

(akj xj )± i =

± akj xj,i if akj 0,

∓ akj xj,i if akj < 0,  ± akj xj,i if akj 0,

∓ akj xj,i if akj < 0.

Introducing the 2n × 2n matrix S whose elements are defined by ⎫ skj = max{0, akj } ⎬ sk,n+j = min{0, akj } k = 1, 2, . . . , n, sn+k,j = sk,n+j ⎭ j = 1, 2, . . . , n sn+k,n+j = skj we can write − + (akj xj )− i = sk,j xj i + sk,n+j xj i , − + (akj xj )+ i = sn+k,j xj i + sn+k,n+j xj i

and if we put ⎡

− x1,i

⎤

⎢ ··· ⎥ ⎥ ⎢ ⎢ − ⎥ ⎢x ⎥ ⎢ n,i ⎥ xi = ⎢ + ⎥ , ⎢ x1,i ⎥ ⎥ ⎢ ⎥ ⎢ · · · ⎦ ⎣ ⎡

+ xn,i − x1,i

⎤

⎢ ··· ⎥ ⎥ ⎢ ⎢ − ⎥ ⎢ x ⎥ ⎢ n,i ⎥ x i = ⎢ + ⎥ , ⎢ x1,i ⎥ ⎥ ⎢ ⎥ ⎢ ⎣ ··· ⎦ + xn,i

⎡

− b1,i

⎤

⎢ ··· ⎥ ⎥ ⎢ ⎢ − ⎥ ⎢b ⎥ ⎢ n,i ⎥ bi = ⎢ + ⎥ , ⎢ b1,i ⎥ ⎥ ⎢ ⎥ ⎢ · · · ⎦ ⎣ + bn,i ⎡ − ⎤ b1,i ⎢ ··· ⎥ ⎥ ⎢ ⎢ − ⎥ ⎢ b ⎥ ⎢ n,i ⎥ bi = ⎢ + ⎥ ⎢ b1,i ⎥ ⎥ ⎢ ⎥ ⎢ ⎣ ··· ⎦ + bn,i

then (35) is solved by 2(N + 1) systems of 2n equations each, for the 4n(N + 1) variables to be determined Sx i = bi , i = 0, 1, . . . , N, S(x i ) = bi , i = 0, 1, . . . , N − + − + under the usual monotonicity conditions for the xk,i , xk,i and xk,i 0 , xk,i
0.  B   A D + B  = A, A − B  = |A| = [|aij |] we have (following [22]) S −1 = [ C As S = [ B A ] with A  ] where  C D  = 1 [A−1 + |A|−1 ] and D  = 1 [A−1 − |A|−1 ]. By separating the lower and the upper components, we get the C 2 2 following form of the solution:  − − + x i = 21 [A−1 + |A|−1 ]bi + 21 [A−1 − |A|−1 ]bi , i = 0, 1, . . . , N, 1 −1 − |A|−1 ]b− + 1 [A−1 + |A|−1 ]b+ x+ i i i = 2 [A 2  − − + x i = 21 [A−1 + |A|−1 ]bi + 21 [A−1 − |A|−1 ]bi , i = 0, 1, . . . , N 1 −1 − |A|−1 ]b− + 1 [A−1 + |A|−1 ]b+ x + = [A i i i 2 2

L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

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and the conditions to have proper fuzzy numbers are − + xj,i
xj,i ,

− xj,i 0,

+ xj,i
0, ∀i, j,

− − xj,i
xj,i+1 ,

∀j, i = 0, 1, . . . , N − 1,

+ + xj,i  xj,i+1 ,

∀j, i = 0, 1, . . . , N − 1.

Examine now the fuzzy linear regression problem with r crisp inputs x1 , x2 , . . . , xr and a fuzzy output Y. The model (see [10,11,20,43] or [4] for general settings and approaches) Y = A0 + x1 A1 + x2 A2 + · · · + xr Ar contains r + 1 fuzzy parameters A0 , A1 , . . . , Ar to be estimated from a set of m observations {Yk , xk,1 , xk,2 , . . . , xk,r |k = 1, 2, . . . , m}. Let − − + + Yk = (yk,i , yk,i , yk,i , yk,i )i=0,1,...,N ,

k = 1, 2, . . . , m

− − + + Aj = (aj,i , aj,i , aj,i , aj,i )i=0,1,...,N ,

j = 0, 1, . . . , r

and

be the LU-fuzzy numbers involved in the model. For this computation we simplify the problem by assuming the positiveness of the observations xk,j 0 ∀j, k; this is not a restriction as we can proceed in a way similar to (35) by splitting the positive and the negative x’s (xk,0 = 1 ∀k). In terms of the LU representation, we can write ( =wi means equality in the weighted least squares sense with weight wi relative to the same values of
=
i ) − − = wi a0,i + yk,i

r

− aj,i xk,j ,

j =1 + + = wi a0,i + yk,i

r

+ aj,i xk,j ,

(36)

j =1 − − = wi a0,i + yk,i

r

− (aj,i )xk,j ,

j =1 + + = wi a0,i yk,i

+

r

+ (aj,i )xk,j

(37)

j =1 − − + + to determine the 4(r + 1)(N + 1) variables aj,i , aj,i , aj,i , aj,i based on the 4m(N + 1) relations (36)–(37). We assume mr + 1. The corresponding least squares estimation can split into three independent constrained linear least squares problems (we omit the details here):   −   − X 0 a y (i) = w with the constraints 0 X a+ y+ − − + + aj,0
· · ·
aj,N
aj,N
· · ·
aj,0 , j = 0, 1, . . . , r

and (ii) {X(a − ) =w y − with the constraints, a − 0, (iii) {X(a + ) =w y + with the constraints, a +
0, where the matrix X has m(N + 1) rows and (r + 1)(N + 1) columns with 1’s, 0’s and xk,j in appropriate locations (we omit the details for brevity).

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Fig. 13. Calculated model (x, Y ) with Y = A0 + A1 x in the D’Urso regression, for different x between 0 and 150. At the values of x corresponding to the m = 6 observations (xk , Yk ), the calculated Y is substituted by the observed Yk .

Their solutions can be obtained by well known existing algorithms (see [33,44]) and software (we use routine DLCLSQ of IMSL) for the constrained linear least squares equations. In the two examples, one is with triangular fuzzy data and the second with nontriangular data. The regression model is with r = 1 Yk = A0 + A1 xk ,

k = 1, . . . , m;

both cases are solved with N = 1, i.e. with values and slopes considered at
= 0 and
= 1; the weights used are w0 = 0.3, w1 = 0.7. In the triangular case, the data are from D’Urso [20] (m = 6) k xk 1 14.8 2 18.0 3 22.9 4 31.5 5 50.3 6 126.0

− − yk,0 yk,0 46.0 13.7 46.0 17.5 48.0 18.8 44.0 26.0 50.0 17.0 56.0 8.2

+ yk,0 78.0 84.0 86.0 90.0 88.0 76.0

+ yk,0 −18.3 −20.5 −19.2 −20.0 −21.0 −11.8

− − yk,1 yk,1 59.7 13.7 63.5 17.5 66.8 18.8 70.0 26.0 67.0 17.0 64.2 8.2

+ yk,1 59.7 63.5 66.8 70.0 67.0 64.2

+ yk,1 −18.3 −20.5 −19.2 −20.0 −21.0 −11.8

The estimated parameters are (A0 is fuzzy triangular and A1 is crisp) A0 = (48.07, 16.87, 83.4, −18.47; 64.93, 16.87, 64.93, −18.47), A1 = (6.07, 0.0, 6.07, 0.0; 6.07, 0.0, 6.07, 0.0) · 10−3 . Fig. 13 reports the observed Yk and the calculated fuzzy function x → A0 + A1 x. For the second example, the nontriangular data are approximated from Kao–Chyu [37] (m = 5) − − yk,0 k xk yk,0 1 2 4.5 0.25 2 5 8.0 0.5 3 7 10.5 0.25 4 10 12.0 0.5 5 12 14.0 0.25

+ + − − + + yk,0 yk,0 yk,1 yk,1 yk,1 yk,1 5.5 −0.25 5.0 5.0 5.0 −5.0 10.0 −0.5 9.0 10.0 9.0 −10.0 11.5 −0.25 11.0 5.0 11.0 −5.0 14.0 −0.5 13.0 10.0 13.0 −10.0 15.0 −0.25 14.5 5.0 14.5 −5.0

The nontriangular (symmetric) estimated fuzzy parameters A0 = (3.185, 0.333, 4.515, −0.333; 3.850, 6.656, 3.850, −6.656), A1 = (0.9187, 0.00239, 0.9283, −0.00239; 0.9236, 0.0478, 0.9236, −0.0478) are represented in Figs. 14 and 15.

L. Stefanini et al. / Fuzzy Sets and Systems 157 (2006) 2423 – 2455

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Fig. 14. Estimated fuzzy parameter A0 in the Kao–Chyu regression model. The membership function is calculated from the LU representation with the mixed spline and N = 1.

Fig. 15. Estimated fuzzy parameter A1 in the Kao–Chyu regression model.The membership function is calculated from the LU representation with the mixed spline and N = 1.

6. Conclusions In this paper we introduce a representation of fuzzy numbers, based on the use of parametrized monotonic functions to model the
-cuts or the membership functions. We call it LU representation, as it models directly and works with the Lower and Upper branches of the fuzzy numbers involved in operations and in fuzzy calculus. The LU-fuzzy numbers can also be viewed as a parametrized extension of standard LR-fuzzy numbers and are related to this extension by a one-to-one correspondence. The paper shows the advantages of use of LU-fuzzy numbers in the principal applications of fuzzy calculus: they generalize the LR-fuzzy setting in the direction of shape preservation but also they allow easy error-controlled approximations in fuzzy calculus. We provide an extensive list of applications of LU-fuzzy numbers and we show the computational advantages associated to their adoption. It appears that they are very flexible and rich in the shapes that are representable and it is extremely easy to implement algorithms of fuzzy calculus that reproduce shape-preserved results. An LU-fuzzy calculator, implementing the basic fuzzy arithmetic operations and calculus by an interactive desk-top easy-to-use software is available by contacting the authors (see [53]). Further work, in preparation, is extending the applications of LU-fuzzy numbers to fuzzy dynamical systems and fuzzy iterated maps (see [55]), to stochastic fuzzy differential equations and to linear fuzzy regression estimation.

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