A generalized and unified approach to the approximation of fuzzy numbers and its arithmetic and characteristics
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Junta de Andalucía Project FQM359
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A generalized and unified approach to the approximation of fuzzy numbers and its arithmetic and characteristics M. I. Berenguer, D. G´amez, A. I. Garralda-Guillem and M. Ruiz Gal´an Department of Applied Mathematics and Institute of Mathematics (IMAG), E.T.S. de Ingenier´ıa de Edificaci´on, University of Granada, Granada, Spain. marib[email protected], [email protected], [email protected], [email protected] Abstract In this paper we propose a general method for the approximation of an arbitrary fuzzy number. This method, which is constructive, recovers and properly extends some well-known approximations such as those obtained in terms of polygonal fuzzy numbers or simple fuzzy numbers. We prove the convergence of the general method and study the properties of the approximation operator, such as its compatibility with arithmetic operations of fuzzy numbers and with some of their important characteristics. In addition to this, we illustrate the method with some particularly interesting cases by providing algorithms, of great simplicity for practical use and apply them to some numerical examples. Furthermore, the approximations we construct are particularly simple from the point of view of fuzzy arithmetic and preserve some of their most important characteristics. 2010 Mathematics Subject Classification: 03E72, 46B15, 65D15. Key words: Fuzzy numbers, Schauder bases, approximation of functions. 1 Introduction The study of fuzzy sets and in particular of the subset of fuzzy numbers has aroused enormous interest in scientific literature in recent decades due to the power that these numbers have to model uncertainty situations in many di↵erent fields of research. The difficulty of extending real models to models with uncertainty using fuzzy numbers, as well as the complexity of fuzzy number calculations, has led to the development of a large number of techniques to approximate arbitrary fuzzy numbers by means of simpler ones. Fuzzy number approximation has been approached over the years from several perspectives, 1 Manuscript Click here to view linked References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
depending on the intended use of the approximation. The introductions of papers [11] and [28] provide a good review of the state of the art. Several papers focus on the approximation of fuzzy numbers by means of simpler ones such as intervals or triangular, trapezoidal, hexagonal... fuzzy numbers, so that both their description and the operations between them are reduced to a finite number of parameters (see [3, 4, 7, 10, 19, 20, 35]). These simplifications, operational from the point of view of fuzzy number computation, present the problem of the loss of information involved in the proposed approximation. To solve this problem, di↵erent modifications of the aforementioned methods have been proposed, preserving some important characteristics of fuzzy numbers. We can mention, without being overly exhaustive, the following: [4, 5, 9, 11, 12, 20, 21, 22, 23, 33, 34, 36]. Some other authors propose approximations by means of polynomial interpolation techniques or splines, attempting to ensure that the interpolant maintains di↵erent properties of shape when combined with the operations of the fuzzy numbers (see [8, 24, 25, 32]). In most of these papers, the initial idea is to fix a finite number of ↵-levels and obtain the best approximation, for a suitable metric, in the set of fuzzy numbers of a certain type associated with these ↵-levels, sometimes preserving some of the characteristics, such as the core, the expected interval, etc. In this paper, however, we remove the initial restriction of fixing a priori the starting ↵-levels. In addition, we construct a sequence of approximating projections which, when applied to a fuzzy number, converges to it for a suitable family of metrics, which includes those classically used. The sequence of approximations thus generated for a fuzzy number is compatible with fuzzy arithmetic as well as with the convergence of the main characteristics associated with that metric. In this paper, we deal with several aspects of fuzzy numbers, all of them related to the approximation of fuzzy numbers by means of computationally easy-to-handle fuzzy numbers, described by simple algorithms and encompassing the two previous perspectives. These ideas, which constitute the contributions of this paper, are developed in more detail below. First, we introduce a family of metrics d(·) Xin the set FXof fuzzy numbers usuch that its lower and upper branches uand uare in a space Xof real functions defined on [0,1]. Such a family encompasses directly, or except equivalences, to the most usual ones, such as the fuzzy Hausdor↵distance ([26]) or the Euclidean distance (see, in essence, [18]). In particular, all the approximation results we obtain for this family of distances apply to all the usual ones. Next, we focus on obtaining the fundamental result, Theorem 4.3, in which we approximate the fuzzy numbers in a metric space (FX,d (·) X) by means of others in that space that are simpler, using as a fundamental tool Schauder bases. These bases have been successfully used in another 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
uncertainty context (see [2]). More precisely, given a fuzzy number u2Fwe construct a sequence of simple fuzzy numbers that always converges to u. Our approach avoids the possible illconditioned problems of some of the best approximation methods that start from a closed convex subset. Moreover, it allows us to approximate any fuzzy number, and establish convergence statements for all types of fuzzy numbers, unlike the results obtained in the above works. For example, when X=Lp[0,1], then FXcoincides with Fand we are able to approximate any fuzzy number by means of another simple fuzzy number, in the sense that it is obtained as a limit of a sequence of simple fuzzy numbers. Moreover, we give an explicit algorithm for the approximation of an arbitrary fuzzy number by means of an arbitrary simple fuzzy number, which is straightforward and for which no additional computation is required, and in which the passage from an nth approximation to the (n+1)th approximation is done by simply adding an additional term consisting of a step fuzzy number, unlike in other works, in which the computation has to be redone. Another application of Theorem 4.3 is given for the set FC[0,1] endowed with the fuzzy Hausdor↵distance, in which a fuzzy number uwith continuous branches, is obtained as the limit of a sequence of very simple fuzzy numbers, the so–called polygonal ones (see [3]). The corresponding algorithm has the same characteristics as those for arbitrary fuzzy numbers. We should also note, with respect to the above algorithms for Fin general and for FC[0,1], that they are compatible with the arithmetic of fuzzy numbers –in particular, with the operations of addition, product by scalars and generalised Hukuhara di↵erence– in the sense that the approximation of the operation coincides with the operation of the approximation. Moreover, we are able to control the distance of a particular operation to its approximation as a function of the elements of the operation and its approximation. Finally, we generalise the convergence results of the approximations obtained in regard to the usual parameters of a fuzzy number, such as value, ambiguity, expected interval and expected value. The paper is structured as follows. In Section 2 we compile the fundamental concepts and results related to the fuzzy numbers that we use. Section 3 focuses on introducing a family of metrics in the set of fuzzy numbers including, among others, the classical Haussdorf distance or the Euclidean distance. Section 4 is devoted to describing the proposed approximation method, including some numerical examples. Moreover, we study the properties of the method in Section 5 and end with some conclusions in Section 6. 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
2 Basic fuzzy concepts We start by recalling the classical concept of a fuzzy number. We should first mention that, in relation to what follows, given a subset Aof a topological space, we write cl (A) for its closure. In addition, we consider in Ronly its usual topology. A fuzzy number (see, e.g., [6]) is a mapping u:R! [0,1] that is i) normal,i.e., there exists x02Rsuch that u(x0) = 1, ii) upper semi-continuous, iii) fuzzy-convex, that is, u(x+(1)y)min{u(x),u(y)},x,y2R,2[0,1], iv) and compactly supported, in the sense that cl {x2R:u(x)>0}is compact. We denote the set of all fuzzy numbers by F. Note that fuzzy-convexity is nothing more than quasi-concavity, a basic notion in other contexts such as convex analysis. We recall an important type of fuzzy number that illustrates the above definition. A fuzzy number u2Fis said to be a simple fuzzy number ([35]) provided that there exist m, n 2, r1,...,r m1,s 1,...,s n12(0,1) and a1,...,a m,b 1,...,b n2Rwith r1<···<r m1,s 1<···<s n1and a1<a mbn<···<b 1, and in such a way that u(x)= 8 > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > : r1,if x2[a1,a 2) r2,if x2[a2,a 3) . . .. . . rm1,if x2[am1,a m) 1,if x2[am,b n] sn1,if x2(bn,b n1] . . .. . . s2,if x2(b3,b 2] s1,if x2(b2,b 1] 0,if x/2[a1,b 1] . A related fuzzy number concept is now given: For a fuzzy number uand a real 0 ↵1, the ↵-level set of u(see, e.g., [6]) is [u]↵:= {x2R:u(x)↵} 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
whenever 0 <↵1, while [u]0:= cl {x2R:u(x)>0}. The 1-level set [u]1={x2R:u(x)1}is called the core of u. The versatility of the level sets is given by this result which, in essence, guarantees that a fuzzy number is uniquely determined by its level sets: Theorem 2.1 ([29]) Let u2Fand for each 0↵1,let[u]↵be its ↵-level set. Then i) for any ↵2[0,1],[u]↵is a closed interval of R,[u]↵=[u↵, u↵], ii) [u]↵2⇢[u]↵1provided that 0↵1↵21, iii) for each sequence {↵n}n1in [0,1] that converges from below to ↵2(0,1] we have that 1 \ n=1 [u]↵n=[u]↵, iv) and for any sequence {↵n}n1in [0,1] that converges from above to 0there holds cl 1 [ n=1 [u]↵n!=[u]0. And conversely, given a family {[u]↵:↵2[0,1]}of subsets of Rfulfilling conditions i) to iv), there exists a unique u2Fsuch that for any ↵2[0,1],[u]↵is its ↵-level set. A typical example of the use of the above result is the following notion: For an m2 and a partition 0 = ↵1<↵ 2<··· <↵ m= 1 of the interval [0,1], a fuzzy number u2Fis a polygonal fuzzy number associated with such a partition ([3]) provided that i=1, . . . , m, ↵i<↵↵i+1 )[u]↵=✓1↵↵i ↵i+1 ↵i◆[u]↵i+↵↵i ↵i+1 ↵i [u]↵i+1 . This concept of polygonal fuzzy numbers includes to that of trapezoidal fuzzy numbers and thus that of triangular fuzzy numbers ([6]). The relationship between fuzzy numbers and the intervals established in Theorem 2.1 allows us to o↵er another representation of a fuzzy number as a pair of functions with some suitable properties. 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Theorem 2.2 ([17]) Let ube a fuzzy number with level sets [u]↵=[u↵, u↵],0↵1.Then, the functions u, u :[0,1] !R, defined at each 0↵1as the endpoints of the ↵-level set [u]↵, u(↵):=u↵,and u(↵):=u↵, satisfy the following properties:: i) uis bounded, non-decreasing, left-continuous in (0,1] and right-continuous at 0, ii) uis a bounded, non-increasing, left-continuous in (0,1] and right-continuous at 0, iii) and u(1) u(1). And conversely, given two functions u, u :[0,1] !Rthat satisfy the above conditions i) to iii), there is a unique fuzzy number u2Fwith uand uas its endpoints of its ↵-level sets [u]↵. Moreover, uis explicitly determined at each x2[0,1] by the expression u(x)=(0,if x/2[u]0 sup{↵2[0,1] : x2[u]↵},if x2[u]0. The representation of a fuzzy number uin terms of the functions uand uis called the LU representation and we refer to uand uas the lower and upper branches of u, respectively. For instance, an equivalent definition of a polygonal fuzzy number in terms of lower and upper branches can be found in [11]. Now we deal with a di↵erent issue: we collect the most important indices related to a fuzzy number, i.e., those real numbers that capture some information contained in a fuzzy number in order to simplify the task of representing and handling it (see [13, 14]). Let u2F: i) If c:[0,1] ! [0,1] is a reducing function, that is, a non-decreasing function with c(0) = 0 and c(1) = 1, then the ambiguity of urelated to cis given by Ambc(u):=Z1 0 c(↵)(u(↵)u(↵)) d↵, while the value of uwith respect to cis the real number Valc(u):=Z1 0 c(↵)(u(↵)+u(↵)) d↵. When, for any ↵2[0,1], c(↵) = 1, we simply write Amb and Val instead of Ambcand Valc, respectively. 6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
ii) The expected interval of the fuzzy number uis the compact interval EI(u):=Z1 0 u(↵)d↵,Z1 0 u(↵)d↵, and the expected value of uis the middle of the previous interval, i.e., EV(u):=1 2Z1 0 (u(↵)+u(↵)) d↵. Thus, the ambiguity of u2Fcan be seen as a measure of its vagueness, its value as a characteristic value of u, its expected interval as an interval containing other significant value of u, its integral, and finally, the expected value of uas a representative value of such an interval. We conclude this section by presenting some elementary aspects of the arithmetic of fuzzy numbers, which essentially reduces them to that of real compact intervals (see, for instance, [6]). So we start with the latter: interval addition, scalar-interval multiplication and (generalized Hukuhara) interval di↵erence. Let A=[a, a] and B=[b, b] be two real compact intervals and 2R. Let us recall that the addition of Aand B, denoted by A+B, is the interval A+B:= [a+b, a +b] and the scalar multiplication of and Ais defined as A:= {a:a2A}=[min{a,a},max{a,a}]. With regard to the interval subtraction, there are several definitions in the literature. One of the most popular is the generalized Hukuhara di↵erence (gH-di↵erence, for short), see [31]. The gH-di↵erence of Aand Bworks like this: A gH B=C() ((a)A=B+C, or (b)B=A+(1)C. Clearly, A gH A=[0,0] and furthermore, the gH-di↵erence of two intervals always exists and A gH B=[min{ab, a b},max{ab, a b}]. We recall the definition of the addition, the scalar-fuzzy number multiplication and the gH-di↵erence of fuzzy numbers. As previosly mentioned, it is a matter of translating interval arithmetic by means of the interval representation of fuzzy numbers given by the level sets. Specifically, if u, v 2Fand 2R, then the addition of uand v,u+v, the scalar-fuzzy number multiplication of and u,u, and, when it exists, the gH-di↵erence of uand v,u gH v, are defined as those fuzzy numbers whose ↵-level sets, for each ↵2[0,1], are, respectively, [u+v]↵:= [u]↵+[v]↵=[u↵+v↵, u↵+v↵], 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
[u]↵:= [u]↵=[min{u↵,u↵},max{u↵,u↵}] and [u gH v]↵:= [u]↵ gH [v]↵=[min{u↵v↵, u↵v↵},max{u↵v↵, u↵v↵}]. Let us recall that the gH-di↵erence of two fuzzy numbers u,v,isthefuzzynumberw, if it exists, such that u gH v=w,((i)u=v+w, or (ii)v=u+(1)w. 3 Metrics in the space of fuzzy numbers Our purpose in this section is to provide a unified treatment of the usual metrics in the space of fuzzy numbers introducing suitable normed spaces. Moreover, we also provide a large family of metrics for Fand even for some relevant subsets of it. In the literature, several metrics are considered in the set Fof fuzzy numbers. The best known and most commonly used metric on that set Fis the Hausdor↵distance (see, e.g. [26, 15]), which is derived from the classical Hausdor↵–Pompeiu distance between compact and convex subsets of Rn, in particular, for compact intervals A=[a, a], B=[b, b]: dH(A, B) := max{|ab|,|ab|}. The fuzzy Hausdor↵distance D1:F⇥F! Ris defined for each u, v 2Fas D1(u, v):= sup ↵2[0,1] max{|u↵v↵|,|u↵u↵|}. Thus, D1(u, v) is a uniform version of dHwhen applied to the level sets of uand v: D1(u, v)= sup ↵2[0,1] {dH([u]↵,[v]↵)}. Other interesting metrics have been introduced in Ffor di↵erent purposes. We highlight, on the one hand, the Euclidean distance d2defined at each u, v 2Fby d2(u, v):=sZ1 0 (u(↵)v(↵))2d↵+Z1 0 (u(↵)v(↵))2d↵, which has been used in [35] to show that the space of simple fuzzy numbers is dense in the space of fuzzy numbers with regard to that metric. Here we also consider the wider family for each 1p<1 dp(u, v):=✓Z1 0 |u(↵)v(↵)|pd↵+Z1 0 |u(↵)v(↵)|pd↵◆1/p , 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
whenever u, v 2F. On the other hand, we also highlight the distance ⇢p:F⇥F! R,1p<1, introduced in [18], where the author introduces a fuzzy number ranking method based on it: ⇢p(u, v) := max (✓Z1 0 |u(↵)v(↵)|pd↵◆1/p ,✓Z1 0 |u(↵)v(↵)|pd↵◆1/p),(u, v 2F). Now we proceed to provide the above-mentioned unified treatment by introducing an appropriate normed space, because the constructive approximation method we will design is based on the use of suitable Schauder bases in certain Banach spaces. Thus, let X⇢R[0,1] and let FX:= {u2F:u, u 2X}. For example, FC[0,1] corresponds to the set of those fuzzy numbers u2Fwith u, u 2C[0,1]. There is an important consideration to be taken into account regarding this set: the fact that afuzzynumberu2Fis continuous has nothing to do with the continuity of uand u, that u2C(R) is independent of u2FC[0,1]. Indeed, if u2FC[0,1] and u(or u) is constant on a proper subinterval of [0,1], then u/2C(R), and a similar argument works in the opposite direction. It should also be noted that for any u2Fwe have that u, u 2L1[0,1]: u, u are measurable because they are monotone, and furthermore, making use of monotonicity again, we have that min{u(0), u(1)}u, u max{u(0),u(1)}. Therefore, FL1[0,1] =F, and since for any 1 p1,L1[0,1] ⇢Lp[0,1], then FLp[0,1] =F. We now consider specific Xsets, real normed spaces of real-valued functions defined on [0,1]. In addition, let k·k (·)be a norm in R2satisfying the monotonicity property 0x1x0 1,0x2x0 2)k(x1,x 2)k(·)k(x0 1,x 0 2)k(·).(3.1) Proposition 3.1 Assume that Xis a real normed space of real-valued functions defined on [0,1], endowed with its norm k·k, and that k·k (·)is a norm in R2fulfilling the monotonicity condition (3.1). Then the mapping d(·) X:FX⇥FX! Rgiven by d(·) X(u, v):=k(kuvk,kuvk)k(·),(u, v 2FX),(3.2) defines a distance in the set FX. 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Proof. In order to establish the validity of a), we check the hypotheses of Theorem 2.2. Thus, let u2FXand n2N. As Pn(u),P n(u)2span {e1,...,e n}and, in view of hypothesis i), the functions e1,...,e nare bounded, left-continuous on (0,1] and right-continuous at 0, so are Pn(u) and Pn(u). Furthermore, Pn(u)(1) Pn(u)(1), since uuis non-increasing and (uu)(1) 0, so, according to our assumption ii), Pn(uu)(1) 0 and the linearity of Pnyields the above inequality. And, in view of iii) and the linearity of Pn, Pn(u) is non-decreasing and Pn(u) is non-increasing. Moreover, we fix u2FXand we must prove that lim n!1 d(·) X(u, Pn(u)) = 0, so, let ">0. The fact that {en}n1is a Schauder basis in Xprovides us with an n02Nsuch that, for all nn0, max {kuPn(u)k,kuPn(u)k}<" k(1,1)k(·), and then d(·) X(u, Pn(u)) = k(kuPn(u)k,kuPn(u)k)k(·) <✓" k(1,1)k(·)," k(1,1)k(·)◆ (·) =" k(1,1)k(·)k(1,1)k(·) =", which all together gives us the proof of b). And finally, given n2N, it is clear that Pnis a projection in FX, according to i) and the fact that Pnis a projection in X. In addition, let Mbe the basic constant of {en}n1.Then kPn(u)Pn(v)kMkuvk and kPn(u)Pn(v)kMkuvk, hence d(·) X(Pn(u),Pn(v)) = ⇣kPn(u)Pn(u)k,kPn(u)Pn(u)k⌘(·) k(Mkuvk,Mkuvk)k(·) Mk(kuvk,kuvk)k(·) =Md(·) X(u, v), which states the M-Lipschitz continuity of Pnmentioned in c). ⇤ 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
In Section 5 we will analyse the properties of the proposed approximations for a given fuzzy number u2FX, the sequence of projections {Pn}n1, their arithmetic and some of their advantages. The next two subsections deal with the respective algorithms to approximate, on the one hand, an arbitrary fuzzy number using a suitable modification of the Haar system, and on the other hand, any fuzzy number in FC[0,1] from the Faber–Schauder system. 4.1 Approximation of an arbitrary fuzzy number We proceed to approximate an arbitrary fuzzy number from Theorem 4.3. As we mentioned in Section 3, given any fuzzy number u2F, its functions u, u belong to Lp[0,1], with p2 [1,1]. Therefore, in the separable case (1 p<1) we can consider the approximations Pn(u) generated from the Haar system, although we must modify that Schauder basis in order that the functions chosen verify the condition i) of Theorem 4.3. As for the Haar system, we fix t0= 0, t1= 1 and for n>1, as n=2 i+kfor some 0 iand some 1 k2i, we take tn=2k1 2i+1 .The modified Haar system consists of those functions on [0,1] that are defined as ˆ h1(t):=1,(0 t1),(4.4) and if n2, then ˆ h2i+1(t):= 8 > > > > > > > > < > > > > > > > > : 1,if 0 t1 2i+1 1,if 1 2i+1 <t2 2i+1 0,otherwise ,(4.5) while for 2 k2i, ˆ h2i+k(t):= 8 > > > > > > > > < > > > > > > > > : 1,if 2k2 2i+1 <t2k1 2i+1 1,if 2k1 2i+1 <t2k 2i+1 0,otherwise .(4.6) It is clear that the slight modification we have made to the original Haar system determines a Schauder basis in Lp[0,1], since each new function ˆ hnis equal almost everywhere to the corresponding hnof the Haar system, but verifying the conditions i), ii) and iii) of Theorem 4.3: 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
i) It is obvious that each basis function ˆ hnis bounded, left-continuous in (0,1] and rightcontinuous at 0. ii) Suppose that g2Lp[0,1] is non-increasing and that g(1) 0. Then, given n2N, according to the description of the projection Pnin (4.3) (it is defined in the same way almost everywhere as for the Haar system), and maintaining its notation, we have that Pn(g)(1) = |In|1ZIn g(t)dt, therefore Pn(g)(1) 0. iii) Let g2Lp[0,1] be a non-decreasing function, n2Nand t, ˜ t2[0,1] with t˜ t. Making use again of the expression of Pnin (4.3), let us initially consider that tand ˜ tare in the same subinterval Ij. In such a case, it is clear that Pn(g)(t)=Pn(g)(˜ t). If, on the other hand, tand ˜ tbelong to di↵erent subintervals, let us say t2Iiand ˜ t2Ik, then i<kand Pn(g)(t)=|Ii|1ZIi g(⇠)d⇠ Pn(g)(˜ t) =|Ik|1ZIk g(⇠)d⇠, since the projections are the average values of the function in the respective intervals. Let us observe that the modified Haar system {ˆ hn}n1is a monotone Schauder basis, since the Haar system is, and so, the generated projections Pnare non-expansive, thanks to Theorem 4.3. We set the modified Haar system {ˆ hn}n1defined in (4.4), (4.5) and (4.6) as the Schauder basis. Therefore, in view of Theorem 4.3, for an arbitrary u2F(let us recall that F=FLp[0,1]), the sequence of approximations {Pn(u)}n1converges to u, which extends the previously stated result in [35], where the convergence of the sequence of approximations is proven only when the fuzzy number is in FC[0,1]. We describe below an easy algorithm to approximate an arbitrary fuzzy number u2F by means of a simple fuzzy number. The distance dfixed in the algorithm can be any of those described previously such as d(·) Lp[0,1] with p2[1,1). Approximation algorithm of an arbitrary fuzzy number 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Input: Functions uand u, dyadic nodes {tn}n0, modified Haar system {ˆ hn}n1, distance d, tolerance ">0. Set a1 Z1 0 u(↵)d↵,b1 Z1 0 u(↵)d↵ Set a2 Z1 0 u(↵)ˆ h2(↵)d↵,b2 Z1 0 u(↵)ˆ h2(↵)d↵ Set P(u) a1ˆ h1+a2ˆ h2,P(u) b1ˆ h1+b2ˆ h2 Set µ1 a1+a2,µ2 a1a2,⌘1 b1+b2,⌘2 b1b2 for i=1,2,... for k=1,2,...,2i Set n 2i+k Set an Z1 0 u(↵)ˆ hn(↵)d↵ Z1 0 ˆ hn(↵)ˆ hn(↵)d↵ bn Z1 0 u(↵)ˆ hn(↵)d↵ Z1 0 ˆ hn(↵)ˆ hn(↵)d↵ for j=n, n 1,...,2k+1 Set µj=µj1,⌘j=⌘j1 end (for) Set µ2k µ2k1an,⌘2k ⌘2k1an Set µ2k1 µ2k1+an,⌘2k1 ⌘2k1+an Set P(u) P(u)+anˆ hn,P(u) P(u)+bnˆ hn Calculate d(Pn(u),u) if d(Pn,u)<" Set i0 i Set k0 k Set N n stop end (for) end (for) Output: i0,k0,N,{aj}N j=1,{bj}N j=1,{µj}N j=1,{⌘j}N j=1,P(u),P(u). Once we have finished the algorithm, we obtain the coefficients {aj}N j=1,{bj}N j=1,of the projections PN(u) and PN(u), respectively, in the modified Haar basis, as well as the coefficients, {µj}N j=1,{⌘j}N j=1, which allow us to rewrite the projections and recover the simple fuzzy number PN(u) explicitly. For this, we note by {t0 j}N+1 j=0 the rearrangement of {tj}N+1 j=0 in increasing order, 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
and so we have that PN(u)(↵)= N X j=1 µj1Ij(↵),P N(u)(↵)= N X j=1 ⌘j1Ij(↵),(↵2[0,1]), where I1=[t0 0,t 0 1] and, for j=2,3,...N,Ij=(t0 j1,t 0 j]. The simple fuzzy number PN(u)(x)is then PN(u)(x)= 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : 0,if x<µ 1 j 2i0+1 ,if µjx<µ j+1,j=1,2,...,2k01 ik0 2i0,if µix<µ i+1,i=2k0,2k0+1,...,N 1 1,if µNx⌘N Nk0i 2i0,if ⌘Ni+1 <x⌘Ni,i=1,2,...,N 2k01 Nj 2i0+1 ,if ⌘Ni+1 <x⌘Ni,j=N2k0,...,N 1 0,if x>⌘ 1 . Let us observe that the simple fuzzy number PN(u) given in Theorem 4.3 is obtained for approximating any fuzzy number u2F, unlike that which can be found in other papers, where some additional conditions are assumed, such as the strict monotonicity of uand u([35]). Example 4.4 Let us consider the fuzzy number u(x)= 8 > > > > > > > > > < > > > > > > > > > : 0,x<0 x2 4,0x<1 1 4,1x<2 3x 45 4,2x3 11 4(x3)2,3<x5 0,x>5 . We approximate ufor the cases N= 4 and N= 16 using the Haar system, we obtain d(2) L2[0,1](u, P4(u)) = 0.204578 and d(2) L2[0,1](u, P16(u)) = 0.0592643. In Figure 3, we show the graphs of u,u,PN(u) and PN(u) for N= 4 and N= 16, as well as the fuzzy number uand the approximations above. 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
uand P4(u)u,u,P4(u) and P4(u) uand P16(u)u,u,P16(u) and P16(u), Figure 3 4.2 Approximation of a fuzzy number in FC[0,1] We now focus on the approximation of a fuzzy number that satisfies an additional condition, its lower and upper branches being continuous, that is, a fuzzy number u2FC[0,1]. Then, we can consider the approximation Pn(u) obtained by means of the Faber–Schauder system {fn}n1 associated with the nodes {tn}n1, since such a Schauder basis clearly satisfies the hypotheses of Theorem 4.3: taking into account that Pn(u) and Pn(u) are the continuous piecewise linear functions that interpolate uand u, respectively, at nodes {ti}n i=1, so that, Pn(u) is a polygonal fuzzy number. In particular, as this Schauder basis is monotone, it follows from Theorem 4.3 the non-expansiveness of each projection Pn. All this is compiled in the following easy algorithm, where the goodness of approximation is measured with the fuzzy Hausdor↵distance d(1) C[0,1], which will be denoted, for the sake of simplicity, by d. Approximation algorithm of a fuzzy number in FC[0,1] 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Input: Functions uand u,{tn}n1dense sequence in [0,1] with t1= 0, and t2= 1, Faber–Schauder system associated with {tn}n1, distance d, tolerance ">0. Set a1 u(t1), b1 u(t1), Set P(u) a1f1,P(u) b1f1 for i=2,3... Set ai u(ti) i1 X j=1 ajfj(ti), bi u(ti) i1 X j=1 bjfj(ti) Set P(u) P(u)(x)+aifi,P(u) P(u)+bifi Calculate d(Pi(u),u) if d(Pi(u),u)<" Set N i stop end (for) Output: N,{aj}N j=1,{bj}N j=1,P(u), P(u). Remark 4.5 When a fuzzy number u2FC[0,1] additionally satisfies that u, u are Lipschitz continuous, with Lipschitz constant Land L, respectively, then clearly kuPn(u)k2Lmax i=2,...,n(titi1) and kuPn(u)k2Lmax i=2,...,n(titi1), so, not only d(1) C[0,1](u, Pn(u)) !0 as n!1, as stated in Theorem 4.3, but also d(1) C[0,1](u, Pn(u)) 2 max{L, L}max i=2,...,n(titi1). Obviously, we could consider other metrics, for instance, d(2) (C[0,1],k·k2)thus recovering [11, Proposition 1]. Example 4.6 Consider a fuzzy number v2FC[0,1] defined as v(x)=(1 2+x2,2x2 0,x2 or 2 x. We use the Faber-Schauder system over the dyadic partition to approximate vconsidering N=4 and 16 and we obtain d(1) C[0,1](v,P4(v)) = 0.214359 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
uand P4(u)u,u,P4(u) and P4(u) uand P16(u)u,u,P16(u) and P16(u), Figure 4 and d(1) C[0,1](v,P16(v)) = 0.0614435. The graphs associated with these approximations are shown in Figure 4. 5 Arithmetic and properties of the approximations In this section, we focus our attention on the study of the compatibility between the approximations obtained in Theorem 4.3 for a fuzzy number and the usual operations of fuzzy arithmetic. We also analyse how such approximations allow us to obtain easy approximations for the ambiguity, value, expected interval and expected value of any fuzzy number. We begin with the first of these issues. The obtained result generalises [35, Theorem 6] and [3, Proposition 21]. Proposition 5.1 With the notation and, under the assumptions of Theorem 4.3, let u, v 2FX, 2Rand n2N. Then, we have that i) Pn(u+v)=Pn(u)+Pn(v) 23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
and d(·) X(u+v,Pn(u+v)) d(·) X(u, Pn(u)) + d(·) X(v,Pn(v)). In particular, Pn(u+)=Pn(u)+ and d(·) X(u+,Pn(u+)) d(·) X(u, Pn(u)). ii) Pn(u)=Pn(u) and d(·) X(u, Pn(u)) = ||d(·) X(u, Pn(u)). iii) Suppose that u gH vexists. Then Pn(u) gH Pn(v)exists, Pn(u) gH Pn(v)=Pn(u gH v) and d(·) X(u gH v,Pn(u gH v)) d(·) X(u, Pn(u)) + d(·) X(v,Pn(v)). Proof. Let u, v 2FX,2Rand n2N. First of all, we deal with the addition. The additivity of Pn,Pn(u+v)=Pn(u)+Pn(v), follows from that of the projection Pn, theorem 4.3 a) and the definition of the interval sum, which equivalently yield ↵2[0,1] )[Pn(u+v)]↵=[Pn(u)]↵+[Pn(v)]↵. As a consequence, and taking into account the monotonicity condition (3.1), we arrive at d(·) X(u+v,Pn(u+v)) = ⇣ku+vPn(u)Pn(v)k,ku+vPn(u)Pn(v)k⌘(·) ⇣kuPn(u)k+kvPn(v)k,kuPn(u)k+kvPn(v)k⌘(·) d(·) X(u, Pn(u)) + d(·) X(v,Pn(v)). The other statement is obvious, since Pn()=. Regarding the scalar-fuzzy number multiplication, Pn(u)=Pn(u), that is, ↵2[0,1] )[Pn(u)]↵=Pn(u)]↵, 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
it is fulfilled, because, due to the homogeneity of Pn, [Pn(u)]↵=[Pn(u)(↵),Pn(u)(↵)] =[min{Pn(u)(↵),Pn(u)(↵)},max{Pn(u)(↵),Pn(u)(↵)}] =[min{Pn(u)(↵),Pn(u)(↵)},max{Pn(u)(↵),Pn(u)(↵)}] =[Pn(u)]↵. And as a result, d(·) X(u, Pn(u)) = ⇣kuPn(u)k,kuPn(u)k⌘(·) =||⇣kuPn(u)k,kuPn(u)k⌘(·) =||d(·) X(u, Pn(u)). And finally, for the gH-di↵erence of the fuzzy numbers uand v, we assume that it exists and write w:= u gH v.Then,ifu=v+w,byi),Pn(u)=Pn(v)+Pn(w), while when v=u+(1)w, i) and ii) imply that Pn(v)=Pn(u)+(1)Pn(w). Therefore, Pn(u) gH Pn(v) exists and Pn(u) gH Pn(v)=Pn(u gH v). To conclude, we prove the validity of the preciously mentioned control of the distance between u gH vand Pn(u gH v), and, as we have just done, we make a distinction according to the form taken by the gH-di↵erence w. Thus, on the one hand, if u=v+w, then, for any ↵2[0,1], there holds [u]↵=[v]↵+[w]↵, and so, [Pn(u)]↵=[Pn(v)]↵+[Pn(w)]↵, which implies w(↵)=u(↵)v(↵) and w(↵)=u(↵)v(↵), and Pn(w)(↵)=Pn(u)(↵)Pn(v)(↵) and Pn(w)(↵)=Pn(u)(↵)Pn(v)(↵), respectively. Hence, d(·) X(u gH v,Pn(u gH v)) = ⇣ku gH vPn(u gH v)k,ku gH vPn(u gH v)k⌘(·) =⇣kuvPn(u) + Pn(v)k,kuvPn(u)+Pn(v)k⌘(·) ⇣kuPn(u)k+kvPn(v)k,kuPn(u)k+kvPn(v)k⌘(·) =⇣kuPn(u)k,kuPn(u)k⌘+⇣kvPn(v)k,kvPn(v)k⌘(·) d(·) X(u, Pn(u)) + d(·) X(v,Pn(v)). 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
[34] Z. Wang, H. Xie, L. Niu, Trapezoidal approximation-preserving the fuzziness of a fuzzy number, Proceedings 2012 9th International Conference on Fuzzy Systems and Knowledge Discovery, FSKD 2012, 191. [35] G. Wang, J. Li, Approximations of fuzzy numbers by step type fuzzy numbers, Fuzzy Sets and Systems 310 (2017), 47–59. [36] C.T. Yeh, Trapezoidal and triangular approximations preserving the expected interval, Fuzzy Sets and Systems 159 (2008), 1345–1353. 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65