Calculations on the Supremum of Fuzzy Numbers Via Lp Metrics

In this paper, it is proved that the supremum of a family of fuzzy number scan be finitely approximated via Lp metrics and the concrete approaches are given. As a byproduct, it is proved that the L1 metric d1 defined via cut-set is equivalent to ametric which can be calculated directly via membership functions. Since the Lp metrics are analytic in nature, the results in this paper may have interesting applications infuzzy analysis. For example, it may provide a method for the computation of various fuzzy-number-valued integrals.DOI : http://dx.doi.org/10.22342/jims.13.2.88.197-208


INTRODUCTION AND PRELIMINARIES
Since the concept of fuzzy number was first introduced in the 1970's, it has been studied extensively from many different viewpoints.Fuzzy numbers has been used as a basic tool in different parts of fuzzy theory.In [4], it is shown that the endograph metric is approximative with respect to orders on E 1 and it is computable.From [3], we know that for uniformly supported bounded set of fuzzy numbers the L p metrics and the endograph metric are equivalent.Thus, we can conclude that L p metrics are also approximative.In [2], a method for calculating supremum and infimum of fuzzy sets via endograph metric is given which resembles the Riemann sum in calculus.In this paper, we will give out the concrete methods to approximate the supremum via L p metrics.
First of all, we recall the basics of fuzzy numbers.Let R and I be the set of all real numbers and the unit interval respectively.
Elements in E 1 are called fuzzy numbers.For α ∈ I, let u α = {x ∈ R|u(x) ≥ α} denote the α − cut of u, then all cuts of u are none-empty closed intervals.For each α ∈ I, let u α = [u − α , u + α ].For any u, vıE 1 , define: where p ≥ 1 is an arbitrary real number.Then d p is a separable but not complete metric on E 1 , called L p metrics.By the definition of L p metrics, we have the following inequalities. where p .We note that in the literature, fuzzy numbers can also be equivalently defined as follows: u = (l u , r u ), where l u and r u are functions defined on certain closed intervals [a u , c u ] and [c u , b u ] with codomain I respectively, such that l u (a u ) = 0, l u (c u ) = 1, r u (c u ) = 1, r u (b u ) = 0; they are increasing and decreasing respectively, and both are upper u. s. c.. For our need we require that the domains of definition for l u and r u are the whole real line R, so that they can be extended uniquely to keep their monotonicity.That is, outside their domains, their values should be either 0 or 1 according to the monotonicity requirement. For α for all α ∈ I, or equivalently, l u ≥ l v and r u ≤ r v .Then ≤ is a partial order on E 1 .
For other undefined notions, we refer to [1].Remark 1 By the definition of order relation on E 1 and the definition of L p metrics, it is obvious that for u, v, w ∈ E 1 , if u ≤ v ≤ w, then d p (u, v) ≤ d p (u, w), i.e., L p metrics preserve the order on fuzzy numbers.

CALCULATIONS VIA L 1 METRIC
In this section, we give some concrete calculation methods to approximate the supremum of fuzzy numbers via L 1 metric.Moreover, we prove that the L 1 metric defined via cut-set can be directly studied by a metric on E 1 which is defined via membership functions and give out the calculation approach on the supremum directly via membership function method.
Proposition 1 [8] Let {u t |t ∈ T } be a family of fuzzy numbers with α − cut representations {(u t ) α |α ∈ [0, 1]}, t ∈ T .If the family is bounded above, and v is the supremum of the family, then the cut-set functions of v have the following representation: for α ∈ (0, 1], where Disc(v + ) is the set of all discontinuous points of v + , which is at most countable; while We The above representation of supremum is based on the cut-set functions.There is also a representation of supremum based on the membership functions as follows: Proposition 2 [4] Let {u t |t ∈ T } be a family of fuzzy numbers.If the family is bounded above, v the supremum of the set, then the membership function of v is given by the following formula: otherwise.
Where Disc + (v) is the set of all discontinuous points of v greater than v − 1 , which is at most countable since v is quasiconvex as a real function.The closure is taken in the induced fuzzy topological space (I R , ω(τ )), where τ is the usual topology on R. (For related concepts on fuzzy topology, we refer to [6] and [7].)Now, we consider the fuzzy set w ′ whose membership function is defined as follows: Note that w ′ (v − 1 ) = 1 and v is the smallest u. s. c. function greater than w ′ .w ′ is u.s. c. if only if w ′ ∈ E 1 , i.e., w ′ = v. w ′ can also be equivalently defined as follows: w ′ = (l w ′ , r w ′ ), the definitions of l w ′ and r w ′ are similar to the case of fuzzy numbers, that is l w ′ (x) = t∈T l ut (x), r w ′ (x) = t∈T r ut (x).By the definition of v, we have l w ′ = l v and r w ′ (x) = r v (x) only if x ∈ Disc + (v).Thus r w ′ and r v differ at most on a countable set.Now, we proceed to consider the relation between w and w ′ .In fact, w does not necessarily correspond to the cut-set function of a fuzzy number, since w + may not be left continuous.In general, w may not even be a cut-set function of a fuzzy set.But we can define a fuzzy set w * on R according to w as follows: The following example shows that w + = v + and w ′ = v in general.
Lemma 1 Assume that {u n } and v are given as in Proposition 1, w is defined as above, and (1) Proof Since d 1 (w, v) = 0, so we only need to show that d 1 (u, w) < 2ε.First, we show that u − − w − 1 < ε.Define two simple functions h 1 and h ′ 1 on [0, 1] as follows: Clearly, by (1) and the monotonicity of w − α and u − α , we have Second, we show that w + − u + 1 < ε.Similar to the above case, we also define two simple functions h 2 and h ′ 2 on [0, 1] as follows: . ., n.By a similar argument as above, we have Based on the above discussion, we have the following algorithm of computing the supremum of fuzzy numbers via the L 1 metric d 1 .
Theorem 1 Under the hypothesis of Proposition 1.
In the following, we consider the case when the family of fuzzy numbers is given by their membership functions.It can be seen that the calculations of approximation with respect to supremum via L 1 metric can be carried out in a similar way as in the cut-set case.By the geometric meaning of integration and the integration variable transformation, we have the following lemma. and Proof Here, we only prove (3).The proof for (4) is similar.In order to prove this lemma, we resort to the Lebesgue measure of real two-dimensional space R 2 .Let .
Here Disc(u − ) and Disc(v − ) are the sets of all discontinuous points of u − and v − respectively.Clearly, A i (i = 1, 2, 3) and B are measurable.The Lebesgue measure of A i (i = 1, 2, 3), A and B are denoted by m(A i )(i = 1, 2, 3), m(A) and m(B), respectively.By the meaning of integration and Lebesgue measure of R 2 , |dx.Now, we show that A = B. First, for each (x, α) ∈ A, there are three cases: Case As l u (x) is upper semicontinuous on R, l u (x) = α.Thus, (x, α) ∈ B. Case 3. (x, α) ∈ A 3 .Similarly to the case 2, we can show that (x, α) ∈ B.
Thus, by the above discussion we have A ⊆ B.
Remark 2 From Lemma 2, it can be seen that the integral defined via cut-set can be represented by the integral defined via corresponding membership functions which is more direct in certain cases.By the properties of l u , l v and r u , r v , we can arbitrarily extend the integration interval of the formulas on the right hand side of ( 3) and ( 4), but their integration values remain the same: Now, we define another metric on E 1 , which is based on the membership functions of fuzzy numbers.For any u, v ∈ E 1 , define Clearly, d * 1 is a metric on E 1 .By the definition of d 1 , we have So by Lemma 2, Remark 2 and the definition of d * 1 , we have the following theorem.

Remark 3
The metric d * 1 on E 1 is based on the membership functions of fuzzy numbers.It is more direct than the L 1 metric d 1 in certain cases since d 1 is based on the cut-set functions.From Theorem 2, we can see that the metrics d 1 and d * 1 are uniformly equivalent.So they have the same topological properties.Hence, the metric d * 1 is also approximate to the supremum and it is computable.
Lemma 3 Assume that {u n } and v are given as in Proposition 1, w is defined as above, and and where Proof By Lemma 2 and Remark 2, we have By the discussion of Lemma 2, we have the following algorithm on the supremum of fuzzy numbers based on membership functions.Theorem 3 Assume that {u t |t ∈ T } and v are given as in Proposition 1, w is defined as above. . .
Remark 4 Note that in the proof of Lemma 3, we only used the values of u at the isolated points: Thus, if we define , and let u ′ be linear between x i and , and u ′ is piecewise linear and hence continuous, thus we have the following: Theorem 4 Under the hypothesis of Proposition 2, for every ǫ > 0 there is a piecewise linear fuzzy number u ′ which is determined by a finite number of values of a finite number of fuzzy numbers of {u t |t ∈ T } such that d 1 (u ′ , v) < 2ε.

CALCULATIONS VIA L p (p ≥ 1) METRICS
In this section, we discuss the finite approximate algorithm for supremum with respect to the general L p metrics.The situation will be much more complicated than the case when p = 1.
. Now we define the partition points x j as follows: Case 1. E ′ j is an interval (the interval is right closed since f is left continuous), take 0. So they form a set of partitioning points for [0, 1].It may happen that {x i |i = 0, 1, • • • n} have repeated points.For our need we can rename these partition points and make them have no repeated points.So 0 = x 0 < x 1 < • • • < x n = 1 when f is increasing and 1 = x 0 > x 1 > • • • > x n = 0 when f is decreasing.
Remark 5 Based on the above partition and by the monotonicity of the function f , we have 0 Lemma 4 Assume that w and and v are given as in Proposition 1, u is a fuzzy number such that u ≤ v. Let u − 0 = a, w − 1 < b.For the two functions w + and w − , pick partitioning points x exists, and it is denoted by w . By the left continuity of w + , we have 0 Based on the above discussion, we have the following algorithm on the supremum of fuzzy numbers.In this paper, we have shown that the supremum of a family of fuzzy numbers can be finitely approximated via L p metrics and give out the concrete approach to approximate the supremum.As our computation method is finite, it might be executed by computers.As a byproduct, it is proved that the L 1 metric d 1 defined via cut-set is equivalent to a metric which can be calculated directly via membership functions.The results in this paper also show that L p metrics are useful metrics on fuzzy number spaces.Because the approximation to the supremum via L p metrics are feasible and computable, moreover, the L p metrics are analytic in nature, so our result may have applications in fuzzy analysis.For example, it could provide a method for the computation of various fuzzy-number-valued integrals.