key: cord-0044800-8co0w8nc
authors: Stefanini, Luciano; Sorini, Laerte; Shahidi, Mina
title: New Results in the Calculus of Fuzzy-Valued Functions Using Mid-Point Representations
date: 2020-05-15
journal: Information Processing and Management of Uncertainty in Knowledge-Based Systems
DOI: 10.1007/978-3-030-50143-3_3
sha: 271e6f01a728836f1d4cc0d0e4f6bccadd5305b4
doc_id: 44800
cord_uid: 8co0w8nc

We present new results in the calculus for fuzzy-valued functions of a single real variable. We adopt extensively the midpoint-radius representation of intervals in the real half-plane and show its usefulness in fuzzy calculus. Concepts related to convergence and limits, continuity, level-wise gH-differentiability of first and second orders have nice and useful midpoint expressions. Using mid-point representation of fuzzy-valued functions, partial orders and properties of monotonicity and convexity are discussed and analysed in detail. Periodicity is easy to represent and identify. Graphical examples and pictures accompany the presentation.

We denote by K C the family of all bounded closed intervals in R, i.e., K C = a − , a + | a − , a + ∈ R and a − ≤ a + .

To describe and represent basic concepts and operations for real intervals, the well-known midpoint-radius representation is very useful (see e.g. [2] and the references therein): for a given interval A = [a − , a + ], define the midpoint a and radius a, respectively, by a = a + + a − 2 and a = a + − a − 2 ,

When we refer to an interval C ∈ K C , its elements are denoted as c − , c + , c, c, with c ≥ 0, c − ≤ c + and the interval by C = [c − , c + ] in extreme-point representation and by C = ( c; c), in midpoint notation.

Given A = [a − , a + ], B = [b − , b + ] ∈ K C and τ ∈ R, we have the following classical (Minkowski-type) addition, scalar multiplication and difference:

Using midpoint notation, the previous operations, for A = ( a; a), B = ( b; b) and τ ∈ R are:

We denote the generalized Hukuhara difference (gH-difference in short) of two intervals A and B as A gH B = C ⇐⇒ (A = B + C or B = A − C); the gH-difference of two intervals always exists and, in midpoint notation, is equal to

The gH-addition for intervals is defined by

If A ∈ K C , we will denote by len(A) = a + − a − = 2 a the length of interval A. Remark that αA − βA = (α + β)A only if αβ ≥ 0 (except for trivial cases) and that A gH B = A − B or A ⊕ gH B = A + B only if A or B are singletons.

For two intervals A, B ∈ K C the Pompeiu-Hausdorff distance d H :

with d(a, B) = min b∈B |a − b|. The following properties are well known:

It is known (see [4, 8] 

It is also well known that (K C , d H ) is a complete metric space. A fuzzy set on R is a mapping u : R → [0, 1]. we denote its α-level set as

A fuzzy set u on R is said to be a fuzzy number if:

Let R F denote the family of fuzzy numbers. So, for any u ∈ R F we have [u] α ∈ K C for all α ∈ [0, 1] and thus the α-levels of a fuzzy number are given by

In midpoint notation, we will write

1 is a singleton then we say that u is a fuzzy number. Triangular fuzzy numbers are a special type of fuzzy numbers which are well determined by three real numbers

It is well known that in terms of α-levels and taking into account the midpoint notation, for every α

The following LgH-difference is somewhat more general than the gHdifference: Definition 1. For given two fuzzy numbers u, ν, the level-wise generalized Hukuhara difference (LgH-difference, for short) of u, ν is defined as the set of interval-valued gH-differences

The LU-fuzzy partial order is well known in the literature. Let us recall that given u, ν ∈ R F and given α ∈ [0, 1], heir α-levels are

Definition 2. [7] Given u,ν ∈ R F and given α ∈ [0, 1], we say that

Correspondingly, the analogous LU -fuzzy orders can be obtained by

The corresponding reverse orders are, respectively, u LU ν ⇐⇒ ν LU u, u LU ν ⇐⇒ ν LU u and u LU ν ⇐⇒ ν ≺ LU u.

Using α-levels midpoint notation u α = ( u α ; u α ), ν α = ( ν α ; ν α ) for all α ∈ [0, 1], the partial orders (a) and (c) above can be expressed for all α ∈ [0, 1] as

; the partial order (b) can be expressed in terms of (LU a ) with the additional requirement that at least one of the inequalities is strict.

In the sequel, the results are expressed without proof because they are similar to the ones in [12] and [13] . For the family of intervals u LgH ν we write u LgH ν LU 0 (and similarly with other orders) to mean that w α LU 0 for all w α ∈ u LgH ν.

We say that u and ν are LU-incomparable if neither u LU ν nor u LU ν and u and ν are α -LU-incomparable if neither u α−LU ν nor u α−LU ν.

The following are equivalent:

In midpoint notation, let [F (x)] α = ( f α (x); f α (x)) and L α = ( l α ; l α ) for all α ∈ [0, 1]; then the limits and continuity can be expressed, respectively, as

and

The following proposition connects limits to the order of fuzzy numbers. Analogous results can be obtained for the reverse partial order LU . 

Similar results as in Propositions 4 and 5 are valid for the left limit with

The graphical representation of a fuzzy-valued function is then possible in terms either of the standard way, by picturing the level curves y = f − α (x) and y = f + α (x) in the plane (x, y), or, in the half-plane ( z; z), by plotting the parametric curves z = f α (x) and z = f α (x); Figs. 1 and 2 give an illustration of the two graphical alternatives for the (periodic, with period 2π) fuzzy function In relation with the LgH-difference, we consider the concept of LgHdifferentiability. 

if F LgH (x 0 ) α is a compact interval for all α ∈ [0, 1], we say that F is levelwise generalized Hukuhara differentiable (LgH-differentiable for short) at x 0 and the family of intervals

Also, one-side derivatives can be considered. The right LgH-derivative of

. The LgHderivative exists at x 0 if and only if the left and right derivatives at x 0 exist and are the same interval.

In terms of midpoint representation [F ( that the midpoint function f α is required to admit the ordinary derivative at x. With respect to the existence of the second limit, the existence of the left and right derivatives f (l)α (x) and

or, in the standard interval notation,

The monotonicity of F : [a, b] → R F can be analyzed also locally, in a neighborhood of an internal point

We omit the corresponding definitions as they are analogous to the previous ones. 

is nonincreasing at x 0 for all α ∈ [0, 1]; (iii) Analogous conditions are valid for strict and strong monotonicity.

In Fig. 4 we picture the fuzzy-valued first order LgH-derivative of a function F (x) and in Fig. 5 we show graphically the membership functions of F LgH (x) at two points x = 2.2 and x = 4.5. Observe from Fig. 5 that F LgH (4.5) is positive in the ( LU ) order and that F LgH (2.2) is negative in the same order relation, denoting that F (x) is locally strictly increasing around x = 4.5 and locally strictly decreasing around x = 2.2 (see also Fig. 4) . 2) and F LgH (4.5) of the fuzzy-valued LgH-derivative of function F (x) described in Fig. 1 . In the midpoint representation, a vertical curve corresponds to the displacement of the n = 11 computed α-cuts; the red lines on the right pictures reconstruct the α-cuts. Remark that y and z represent the same domain and that a linear vertical segment in the midpoint representation corresponds to a symmetric membership function. (Color figure online) Analogous results are also immediate, relating strong (local) monotonicity of F to the "sign" of its left and right derivatives F (l)LgH (x) and F (r)LgH (x); at the extreme points of [a, b], we consider only right (at a) or left (at b) monotonicity and right or left derivatives.

for all α ∈ [0, 1] with left and/or right gH-derivatives at a point Fig. 6 . Level-wise endpoint graphical representation of the fuzzy-valued LgH-derivative

The core is intercepted by the black-colored curves. The other α-cuts are represented by red-colored curves for the left extreme functions f − α (x) and blue-colored curves for the right extreme functions f + α (x). (Color figure online)

We have three types of convexity, similar to the monotonicity and local extremum concepts.

Let F : [a, b] → R F be a function and let LU be a partial order on R F . We say that 

In Fig. 6 we picture the fuzzy-valued second order LgH-derivative of a function F (x) and in Fig. 7 we show graphically the membership functions of F LgH (x) at two points x = 2.2 and x = 4.5. Observe from Fig. 7 that F LgH (2.2) and F LgH (4.5) are not positive nor negative in the ( LU ) order, denoting that F (x) is (locally) not convex nor concave around x = 2.2 or x = 4.5. See also Fig. 6 where regions of positive and negative second-order derivative can be identified.

Analogously to the relationship between the sign of second derivative and convexity for ordinary functions, we can establish conditions for convexity of fuzzy functions and the sign of the second order LgH-derivative F LgH (x); for example, a sufficient condition for strong ≺ LU -convexity is the following (compare with Proposition 10): Obviously, if F has a period T , then this also implies that F α for all α ∈ [0, 1] has a period T i.e., for all α ∈ [0, 1], f α and f α are periodic with period T . On the other hand, the periodicity of functions F α for all α ∈ [0, 1] does not necessarily imply the periodicity of F . (1) if the periods T and T are commensurable, i.e., T T ∈ Q ( T T = p q , such that p and q are coprime) then the function F is periodic of period T = lcm( T , T ), i.e,. T is the least common multiple between T and T (i.e., T = p T = q T );

(2) if the periods T and T are not commensurable, i.e., T T / ∈ Q, then function F is not periodic.

A periodic function of period p = 2π is given in Fig. 8 . It is the function reported in all figures above, extended to the domain [0, 6π].

We have developed new results to define monotonicity ans convexity or concavity for fuzzy-valued functions, in terms of the LU-partial order; similar results can be obtained for other types of partial orders. It appears that midpoint representation of the α-cuts is a useful tool to analyse and visualize properties of fuzzy-valued functions. In further work, we will analyse important concepts of (local) minimal and maximal points for fuzzy valued functions, by the use of first-order and second-order LgH-derivatives.

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