Solutions of fuzzy equations based on Kaucher arithmetic and AE-solution sets

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1 Fuzzy Sets and Systems 159 (2008) Solutions of fuzzy equations based on Kaucher arithmetic and AE-solution sets T. Rzeżuchowski,J.W asowski Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, Warsaw, Poland Received 7 May 2007; received in revised form 7 December 2007; accepted 22 January 2008 Available online 17 February 2008 Abstract We introduce and investigate two new types of fuzzy solutions to the equation f(a,x)= b where the parameters a and b are described in a fuzzy way. The solutions we propose are based on some known solutions to this type of equation with the parameters described by the interval data. We adapt to the fuzzy case so-called formal interval solutions (based on the Kaucher interval arithmetic) and AE-solution sets. In order to define fuzzy formal solutions we introduce a definition of fuzzy Kaucher arithmetic. We investigate the regularity of solutions in the case of linear dependence of f on a and discuss also the problem of interpretation and logical description of formal solutions Elsevier B.V. All rights reserved. MSC: 03E72; 08A72 Keywords: Fuzzy numbers; Algebra; Kaucher arithmetic; Algebraic equations; AE-solution sets 1. Introduction and some basics of fuzzy sets Let f : R m R n R s, f = (f 1,...,f s ) (1) and consider the equation f(a,x)= b, (2) where a = (a 1,...,a m ) R m, x = (x 1,...,x n ) R n. Our main interest in this paper concerns the situation which occurs when the data a and b for that equation are not exactly known and the information is available under the form of fuzzy sets. We propose to extend Kaucher arithmetic used in the interval mathematics to the case of fuzzy sets and we use it to define solutions to Eq. (2). We adapt also to the fuzzy situation so-called AE-solution 1 sets known in the theory of This research was supported by a grant from the Faculty of Mathematics and Information Science of Warsaw University of Technology. Corresponding author. Tel.: ; fax: addresses: tarz@mini.pw.edu.pl (T. Rzeżuchowski), januszwa@mini.pw.edu.pl (J. W asowski). 1 The letters AE are used due to their resemblance to the for all and exists quantifiers the reasons for this are explained in Section /$ - see front matter 2008 Elsevier B.V. All rights reserved. doi: /j.fss

2 T. Rzeżuchowski, J. W asowski / Fuzzy Sets and Systems 159 (2008) equations with interval data (see [19]) and give some properties of both kinds of solutions in the fuzzy setting. The solutions are defined as families of solutions of parametrized sets of equations with interval data note that they are different than considered in the literature so-called equations with interval parameters (see [16]). By F(R n ) we denote the class of all fuzzy sets on R n by which are meant functions defined on R n with values in [0, 1]. Fuzzy sets and other fuzzy objects built of them will be denoted by letters like Ã, x with a tilde. In Section 3.1 (Definitions 3.1 and 3.2), we introduce a generalization of this structure. Its elements will be denoted like  or ˆx. The function defining a fuzzy set is often called its membership function and is denoted as Ã( ). Fuzzy sets are described also by the family of so-called α-cuts [Ã] α ={x R n Ã(x) α} for α (0, 1] and [Ã] 0 defined separately as the support of Ã( ). We distinguish the class FR(R n ) of regular fuzzy sets satisfying the following three conditions: (i) à is normal, i.e. there exists x 0 R n with Ã(x 0 ) = 1. (ii) The membership function Ã( ) is upper semicontinuous. 2 (iii) [Ã] 0 is a compact subset of R n. In FR(R) we distinguish the subclass FI of fuzzy intervals, i.e. à for which all the α-cuts are intervals. If in addition there is exactly one real number u such that Ã(u) = 1 then à is called a fuzzy number. A real number a is identified with ã for which ã(a) = 1 and ã(x) = 0ifx = a. Regular fuzzy sets are characterized by the properties of families of their α-cuts this is summarized in the following theorem (see [15]). Theorem 1.1. For every fuzzy set à FR(R n ) the following conditions are satisfied: (1) For all α [0, 1] the α-cuts [Ã] α are compact in R n. (2) If 0 α β 1 then [Ã] β [Ã] α. (3) If α k (0, 1] and α k α then [Ã] α = k 1 [Ã]α k. Conversely, if a family {K α 0 α 1}, with K 0 = cl( {K α 0 < α 1}), satisfies the above three properties ([Ã] α is replaced with K α ) then there is à FR(R n ) such that [Ã] α = K α for 0 < α 1. Moreover, Ã(u) = sup{α u K α } for u K 0 and is equal to 0 outside K 0. A fuzzy interval à may be represented in an equivalent way by a pair of functions (see [9]). In fact, as each [Ã] α is an interval, we may write [Ã] α =[A l (α), A r (α)] and A l,a r become functions defined on [0, 1]. Their properties are gathered in the following theorem. Theorem 1.2. For every fuzzy interval à FI: (1) A l is a bounded, left continuous, nondecreasing function on (0, 1]. (2) A r is a bounded, left continuous, nonincreasing function on (0, 1]. (3) A l and A r are both right continuous at 0. (4) A l (1) A r (1). Conversely, if a pair of functions u l,u r : [0, 1] R satisfies the above four properties, with A l and A r replaced respectively by u l and u r, then there exists a unique à FI for which [Ã] α = [u l (α), u r (α)] for α [0, 1]. Let now L, R :[0, 1] [0, 1] be nonincreasing and left continuous functions such that L(0) = R(0) = 1 and L(x), R(x) (0, 1) for x (0, 1). For any four reals a l,a r, α l, α r such that a l a r and α l, α r > 0 we may define a 2 Recall that upper semicontinuity of Ã( ) means that for every x 0 in its domain lim sup x x0 Ã(x) Ã(x 0 ).

3 2118 T. Rzeżuchowski, J. W asowski / Fuzzy Sets and Systems 159 (2008) fuzzy interval à using the following membership function: 1 if a l x a r, L((a Ã(x) = l x)/α l ) if a l α l x<a l, R((x a r )/α r ) if a r <x a r + α r, 0 otherwise. à is then called L R fuzzy interval, usually denoted as à = (a l,a r, α l, α r ) LR. The interval [a l,a r ] is called kernel of Ã, α l and α r its left and right spreads. Note that [Ã] α =[a l α l L ( 1) (α), a r + α r R ( 1) (α)], where L ( 1), R ( 1) :[0, 1] [0, 1] are quasi-inverses of L, R defined by L ( 1) (α) = sup{x [0, 1] L(x) α}, R ( 1) (α) = sup{x [0, 1] R(x) α}. For fuzzy numbers we use a shortly abbreviated notation à = (a, α l, α r ) LR the kernel is reduced here to a singleton. Particularly important examples of fuzzy intervals are provided by so-called piecewise linear fuzzy intervals where L(x) = R(x) = 1 x for all x [0, 1]. In this case the subscript LR will be omitted in the notation we simply write à = (a l,a r, α l, α r ). 2. Equations under interval type uncertainty This section is a toolbox for the next one where we investigate equations under fuzzy uncertainty. First we recall the basic notions of Kaucher interval arithmetic and next show how using this arithmetic formal solutions of (2) are defined. At the end we give some basic information concerning so-called AE-solution sets Kaucher interval arithmetic By IR we denote the set of all nonempty, closed, bounded intervals in R identified with ordered pairs of numbers [x, x] where x x. Algebraic properties of classical interval arithmetic defined on IR (see [14]) are often insufficient if we want to deal with equations: intervals with nonzero width do not have inverses in IR with respect to the arithmetical operations. The incompleteness of that algebraic structure we mentioned above stimulated attempts to create a more convenient interval arithmetic extending that based on IR. The one that seems to have been the most successful was created in the 70s by Kaucher [11]. The joint algebraic and order completion of IR carried out in that paper resulted in an algebraic system called now Kaucher interval arithmetic or complete interval arithmetic we describe it very briefly below. We follow the notation from [19]. By KR we denote the set of all ordered pairs of real numbers [x, x] where condition x x is no more required. Thus KR is obtained by adjoining improper intervals [x, x], with x > x, to the set IR of proper intervals. Elements of KR and other interval objects (vectors, matrices with interval entries, etc.) will be represented by boldface characters like A, B, C,...,x, y, z.ifxbelongs to KR then it is a pair and may be denoted as x =[x, x] with the first element x and the second one x. However, the under and overscored letters are not reserved for the first and second elements of pairs, respectively. If we apply so-called dualization mapping dual : KR KR dual [x, x] =[x, x] we get a pair whose first element is x and the second x. Kaucher interval arithmetic is an algebraic structure KR, +,,,/, where the four laws are defined by the formulae below (see [11,19]). Let x =[x, x], y =[y, y] and recall that u + = max{u, 0}, u = max{ u, 0}. x + y =[x + y, x + y], (3) x y =[x y, x y], (4) x y =[max{x + y +, x y } max{x + y, x y + }, max{x y, x + y + } max{x y +, x + y }], (5) x/y = x [1/y, 1/y], (6) where the division in (6) is defined only when yy > 0. Remark that the formulae in (3), (4) and (6) look the same as in the classical interval arithmetic.

4 T. Rzeżuchowski, J. W asowski / Fuzzy Sets and Systems 159 (2008) The arithmetical operations with vectors and matrices made up of elements from KR are defined in a similar way to those known in the classical interval arithmetic (see [19] for the details). The operation dual on interval vectors and matrices is done componentwise. KR is a commutative group with respect to the addition. Multiplication is commutative and associative and if we consider the subfamily of pairs [x, x] for which xx > 0 we get a multiplicative group. Let x =[x, x] KR. Notice that [ x, x], denoted by opp x, is the additive inverse of x and [1/x, 1/x] its multiplicative inverse (the latter only for x x > 0). However, one should be careful with computations in Kaucher arithmetic as, for example, the multiplication is not in general distributive with respect to the addition Formal interval solutions In order to define formal interval solutions we impose on the function f defined in (1) a condition which permits in a natural way to consider its extension to a map defined on KR m KR n. Condition A. The components f i (a, x) (i = 1,...,s) of f are rational expressions, i.e. finite combinations of variables a 1,...,a m,x 1,...,x n and some constants, using elementary arithmetical operations. We consider a natural interval extension of f by which we mean a map F f : KR m KR n KR s (7) generated in the following way: at each f i we replace the real variables a 1,...,a m and x 1,...,x n by interval variables and replace the real arithmetical operations by corresponding interval arithmetical operations defined by (3), (4), (5), (6) (compare with [14]). The only limitation for this procedure is the requirement that if there is a division in the formulae defining f then the divisors which are put directly into f, or which are results of some operations within the formulae defining f, must satisfy the condition yy > 0. Definition 2.1. By a formal interval solution of Eq. (2) with interval parameters a and b at the place of a and b we mean any solution x KR n of the equation F f (a, x) = b. See [19] for the interpretation of solutions in the case when they contain improper intervals it is shown that they have then some real interpretation AE-solution sets We consider an input output system described by the map (1) where some input variables within a = (a 1,...,a m ) R m and the inner state variables x = (x 1,...,x n ) R n play various roles and the roles of output variables by which we mean b k = f k (a, x) are also differentiated in the way described below. The values of parameters a and b are subject to some interval bounds. We recall below the notion of AE-solution sets to (2) with interval parameters a IR m and b IR s (see [19]). With each parameter a i and b k of the system (2) we associate either the quantifier or. This connection will be described in the following way. Let Γ 0 Γ ={1,...,m} and Δ 0 Δ ={1,...,s} (we do not exclude the case Γ 0 = or Δ 0 = ). We define vectors composed of quantifiers A = (A 1,...,A m ) and E = (E 1,...,E s ) where A i = { for i Γ0, for i Γ \ Γ 0, E i = { for i Δ0, for i Δ \ Δ 0. (8) These vectors A, E define the decompositions of interval vectors a and b: a = a + a, b = b + b, (9)

5 2120 T. Rzeżuchowski, J. W asowski / Fuzzy Sets and Systems 159 (2008) where the interval vectors a = (a1,...,a m ), a = (a1,...,a m ), b = (b 1,...,b s ), b = (b 1,...,b s ) are defined below { { ai ai for i Γ = 0, a ai for i Γ \ Γ 0 for i Γ \ Γ 0, i = 0, 0 for i Γ 0, b i = { bi for i Δ 0, 0 for i Δ \ Δ 0, b i = { bi for i Δ \ Δ 0, 0 for i Δ 0. Definition 2.2. The set {x R n a a, b b, a a, b b,f(a + a,x)= b + b } is called interval AE-solution set of AE type to Eq. (2) with interval parameters a and b. It is denoted as Ξ AE (f, a, b). Remark that the interval AE-solution set may be written under the form Ξ AE (f, a, b) = a a b b a a b b {x R n f(a + a,x)= b + b }. (10) An interpretation of interval AE-solutions from the point of view of control theory can be found in [19]. Different decompositions (9) of a and b generate different types of interval AE-solution sets. The three ones shown in the table are most often used. Decomposition (9) The name of AE-solution set Notation a = a, b = b United solution set Ξ uni (f, a, b) a = a, b = b Tolerable solution set Ξ tol (f, a, b) a = a, b = b Controllable solution set Ξ ctr (f, a, b) Even in simple situations arising in practice the direct computation and description of sets Ξ AE (f, a, b) proves, as a rule, arduous and sometimes practically impossible. So it makes sense to try their approximate description by simpler sets in the form of axis-aligned boxes (interval vectors). This may be an inner or outer approximation (see [19]). Consider now a special case of interval AE-solution sets to linear systems Ax = b, (11) where A = (a ij ) R s,n, b R s and x R n. We admit that there is an uncertainty about the parameters and it is described by an interval (s n)-matrix A = (a ij ) and an interval s-vector b = (b i ). We retain the decomposition of vector b defined in (9) and based on the set Δ 0 Δ ={1,...,s}. To describe the same formalism for the parameters a ij let Γ 0 Γ ={(i, j) 1 i s, 1 j n} and let A = (A ij ) be an (s n)-matrix composed of quantifiers defined by { for (i, j) Γ0, A ij = for (i, j) Γ \ Γ 0. This matrix A defines the following decomposition of A: A = A + A, where the interval (s n) matrices A = (aij ) and A = (aij ) are defined as follows: a ij = { aij for (i, j) Γ 0, 0 for (i, j) Γ \ Γ 0, a ij = { aij for (i, j) Γ \ Γ 0, 0 for (i, j) Γ 0. In view of the general Definition 2.2 the interval AE-solution set of Eq. (11) with interval parameters A and b is given by Ξ AE (A, b) ={x R n A A, b b, A A, b b,(a + A )x = b + b }.

6 3. Equations under fuzzy uncertainty T. Rzeżuchowski, J. W asowski / Fuzzy Sets and Systems 159 (2008) We consider now Eq. (2) and the situation which arises when the values of parameters have fuzzy uncertainties. It means that instead of the vectors of parameters a = (a 1,...,a m ) and b = (b 1,...,b s ) we must use fuzzy vectors à = (à 1,...,à m ) and B = ( B 1,..., B s ) composed of fuzzy intervals. Eq. (2) with fuzzy parameters à and B was discussed in several papers, see, e.g., [1 7,10,12,17,20 22]. A few types of solutions for such equation have been proposed including some based on α-cuts [4]. In this section we introduce two new notions of solutions in the fuzzy case. We start with the definition of Kaucher fuzzy arithmetic which makes it possible to use inverse elements to fuzzy intervals with respect to the addition and multiplication Kaucher fuzzy arithmetic It is known that non-crisp fuzzy numbers do not have inverses with respect to the addition and the multiplication based on Zadeh s extension principle. 3 In order to deal with fuzzy solutions of equations, and in the future also with differential equations, it would be convenient to extend the family of fuzzy intervals and define operations so that the inverses exist. We present below a proposal for such an algebraic structure based on an extension of the notion of fuzzy numbers. The elements of this extended family containing all fuzzy numbers as a subset will be denoted using a hat like â. Definition 3.1. We denote by FKRthe set of all pairs (a l,a r ) of functions a l,a r :[0, 1] R bounded, left continuous on (0, 1] and right continuous at 0. If â = (a l,a r ), ˆb = (b l,b r ) FKR then for each α [0, 1] the pairs [a l (α), a r (α)], [b l (α), b r (α)] belong to KR and so we may apply any operation {+,,,/} of Kaucher arithmetic to get [c l (α), c r (α)] =[a l (α), a r (α)] [b l (α), b r (α)] (12) for α [0, 1] (in the case of division we must assume that b l (α) b r (α) >0 for all α). Two functions c l, c r :[0, 1] R are defined in that way. Lemma 3.1. If â = (a l,a r ), ˆb = (b l,b r ) belong to FKR then ĉ = (c l,c r ) defined by (12) also belongs to FKR for {+,, }. If in addition C >0, α [0, 1], max{b l (α), b r (α)} C or min{b l (α), b r (α)} C (13) then ĉ belongs to FKR also for =/. Proof. We see directly from (3) (6) that c l and c r are left continuous in (0, 1] and right continuous at 0. It is also evident that for {+,, } they are bounded. The boundedness in the case =/ is due to assumption (13). This lemma permits us to formulate the following definition of Kaucher fuzzy arithmetic. Definition 3.2. By Kaucher fuzzy arithmetic we understand the algebraic structure FKR, +,,,/, where +,,,/ are defined by (12) with the additional condition (13) on ˆb in the case of â/ˆb. The following proposition describes the situation connected with the existence of inverse elements with respect to addition and multiplication in Kaucher fuzzy arithmetic their existence is the main reason to introduce this arithmetic. 3 Zadeh s extension principle states that a binary operation :R 2 R can be extended to fuzzy numbers the following way (the convention sup =0is adopted here) (à B)(z) = sup min{ã(x), B(y)}, z R, x y=z where Ã, and B are fuzzy intervals.

7 2122 T. Rzeżuchowski, J. W asowski / Fuzzy Sets and Systems 159 (2008) Proposition 3.1. Every element â FKR has a unique inverse with respect to addition in Kaucher fuzzy arithmetic. An element ˆb = (b l,b r ) FKR has an inverse ˆb 1 with respect to the multiplication in Kaucher fuzzy arithmetic if and only if it satisfies condition (13). The inverse element, if it exists, is unique. The proof is an immediate consequence of (3), (6) and Lemma 3.1. Obviously FI FKR. Remark that if â 1 = (u l,u r ) is the inverse element to â FI within the Kaucher fuzzy arithmetic with respect to {+, } then {dual [u l (α), u r (α)] 0 α 1} FI Formal fuzzy solutions We assume that f satisfies Condition A. The tool which will serve us to define solutions of (2) in that situation will be the following one-parameter family of interval equations: F f ([Ã] α, x) =[ B] α, 0 α 1, (14) where [Ã] α = ([à 1 ] α,...,[ã m ] α ), [ B] α = ([ B 1 ] α,...,[ B s ] α ). (15) Definition 3.3. By a formal fuzzy solution of Eq. (2) with fuzzy parameters à and B at the place of a and b we mean any ˆX ={X α 0 α 1} FKR n such that for every α [0, 1] the interval vector X α = (X1 α,...,xα n ) is a formal interval solution of (2) with interval parameters [Ã] α and [ B] α. By Definition 2.1 this means that X α is a solution to (14) for 0 α 1. This kind of solutions to (2) with fuzzy parameters à and B when in the formulae describing f is used classical interval arithmetic was proposed by Buckley [4] and investigated by many authors. However, the method of solving provides often solutions which do not belong to FI n and they are then rejected (except for the equations considered in [7]). It occurs that such false solutions may have interesting interpretations if treated properly. Remark 3.1. Consider Eq. (2) with fuzzy parameters à and B at the place of a and b.if ˆX FI n and f([ã] α,x α ) = [ B] α for α [0, 1] then ˆX is a solution of considered equation which is based on Zadeh s Extension Principle (see [4]). We consider now a simple but important example of linear equations f(a,x)= a x = b (16) for {+, }, with parameters a and b modelled by Ã, B FI. The interpretation of solutions will be important in the sequel. If à has the inverse element û = (u l,u r ) FKR with respect to (which is always true for =+), the formal solution ˆX can be given the following form: ˆX = û B ={[u l (α), u r (α)] [ B] α 0 α 1}, (17) where denotes either + or in the appropriate space FKR or KR. The intervals [u l (α), u r (α)] are always improper, apart the special case when [Ã] α is a singleton and then also [u l (α), u r (α)] is a singleton. The intervals X α may be proper or improper and it is shown in [18] that both cases can be described in the following way: (I) Case of X α proper X α ={z a [Ã] α, b [ B] α ; a z = b} =Ξ tol (f, [Ã] α, [ B] α ). (18) (II) Case of X α improper dual X α ={z b [ B] α, a [Ã] α ; a z = b} =Ξ ctr (f, [Ã] α, [ B] α ). (19)

8 T. Rzeżuchowski, J. W asowski / Fuzzy Sets and Systems 159 (2008) In view of (18) and (19) we propose to accept a solution ˆX in the case when either ˆX FI or dual ˆX ={dual X α 0 α 1} FI. In other cases (when for some α we have [ ˆX] α in IR and for some other in KR \ IR) the solution ˆX should not be automatically rejected. One may consider that in such a case the intervals à and B are not modelling in a proper way the parameters a and b from the point of view of fuzzy sets theory. It is an open question then whether such modelling is possible within an acceptable class of fuzzy intervals. In general making corrections to à and B we may try to improve the regularity of interval solutions with respect to α Fuzzy AE-solution sets We consider the same vectors composed of quantifiers (8) as defined in Section 2.3 and in the same way we associate with each of parameters a i and b i appearing in (2) a quantifier or. For each α [0, 1] we may use the interval AE-solution sets Ξ AE (f, [Ã] α, [ B] α ) described in Definition 2.2 with [Ã] α and [ B] α being the α-cuts of fuzzy vectors à and B given by (15). This permits us to formulate the following definition of fuzzy AE-solution sets. Definition 3.4. We call the family of sets in R n ˆΞ AE (f, Ã, B) ={Ξ AE (f, [Ã] α, [ B] α ) 0 α 1} a fuzzy AE-solution set of AE type of Eq. (2) with fuzzy parameters à and B. 4. Fuzzy equations linear with respect to parameters 4.1. Single algebraic equations We consider in this section the equation f(a,x)= b, where f : R m R R has the following form f(a,x)= a 1 x + +a m x m. Note that if a 1,...,a m > 0 and b>0 then this equation has a unique solution for x>0 and this solution is a continuous function of parameters a 1,...,a m,b. We say that an element of FKR is positive if both defining it functions (see Definition 3.1) have positive values for all α [0, 1]. Throughout this section a and b are modelled by positive fuzzy parameters à = (à 1,...,à m ) FI m and B FI so if we denote [à i ] α =[a α i, aα i ] and [ B] α =[b α, b α ] then 0 < a α i aα i,0< bα b α. According to Definition 3.3 in order to find formal fuzzy solutions of (20) we should construct, using the natural interval extension F f, a family of corresponding equations (14) and next look for their formal interval solutions. For any fixed α [0, 1] such equation takes shape [a α 1, aα 1 ] [xα, x α ]+ +[a α m, aα m ] [xα, x α ] m =[b α, b α ], (21) where + and are defined by (3), (5) and, of course, [x, x] k =[x, x]... [x, x] k-times. We are interested in positive fuzzy formal solutions of (20) so all x α, x α which may occur in (21) are strictly positive (although we admit the inequality x α > x α ). In that case [x, x] k =[x k, x k ] and taking also into account that all used fuzzy parameters are positive Eq. (21) is equivalent to the system of two independent algebraic equations a α 1 xα + +a α m (xα ) m = b α, (20) (22) a α 1 xα + +a α m (xα ) m = b α. Let x α and x α be positive solutions of the above equations. We denote X α =[x α, x α ]. (23) Proposition 4.1. Consider Eq. (20) with positive fuzzy parameters à FI m and B FI. The family ˆX ={X α 0 α 1}

9 2124 T. Rzeżuchowski, J. W asowski / Fuzzy Sets and Systems 159 (2008) is the only positive, formal fuzzy solution of this equation. Moreover, for any α [0, 1] we have X α = Ξ tol (f, [Ã] α, [ B] α ) if X α is proper, (24) dual X α = Ξ ctr (f, [Ã] α, [ B] α ) if X α isimproper. (25) Proof. The solutions x α and x α as functions of α satisfy the conditions required in Definition 3.1 and so define an element of FKR being a unique, positive formal fuzzy solution of (20). Let α [0, 1]. Propositions 6.2 and 6.3 in [19] imply that if Eq. (21) has a formal solution X α =[x α, x α ] then the following inclusions hold: X α Ξ tol (f, [Ã] α, [ B] α ) if X α is proper, dual X α Ξ ctr (f, [Ã] α, [ B] α ) if X α is improper. It occurs that if the parameters and solutions of (21) are positive then the above inclusions become equalities. In fact, let X α be a proper interval. Then for each ˇb [b α, b α ] we have ˇb b α = a α 1 xα + +a α m (xα ) m < a α 1 u + +aα m um for u>x α, ˇb b α = a α 1 xα + +a α m (xα ) m > a α 1 v + +aα m vm for v<x α. So we have that if v<x α and u>x α then v and u do not belong to Ξ tol (f, [Ã] α, [ B] α ) and thus (24) holds. The proof of (25) is similar. Remark 4.1. If the solution ˆX is a fuzzy interval then it is a solution of considered equation based on Zadeh s Extension Principle (see Remark 3.1). Solutions x α and x α as functions of α satisfy the conditions required in Definition 3.1 and so define a positive element of FKR being a unique positive formal fuzzy solution of (20). We show now that under some additional conditions this formal solution is represented by monotone functions. Fix two functions L and R like in Section 1 and consider fuzzy numbers à i = (a i, α l,i, α r,i ) LR for i = 1,...,m and B = (b, β l, β r ) LR, (26) where a i α l,i > 0 and b β l > 0. We denote Φ 1 (x) = mi=1 a i x i b mi=1 α l,i x i β l, Φ 2 (x) = mi=1 a i x i b β r m i=1 α r,i x i. The properties of solutions x α and x α depend on the position of x1 = x0 and x3 = x0 with respect to x2 = x1 = x 1. Let us underline that x1, x 2 and x 3 may appear in any order. We consider four cases: The case x1 = x 2.WehaveΦ 1(x1 ) = 1 and Φ 1(x2 ) = 0. If x 1 <x 2 then Φ 1 is strictly decreasing on [x1,x 2 ].If x2 <x 1 then Φ 1 is strictly increasing on [x2,x 1 ].WehaveΦ 1(x α ) = L ( 1) (α) for α [0, 1]. The case x2 = x 3.WehaveΦ 2(x2 ) = 0 and Φ 2(x3 ) = 1. If x 2 <x 3 then Φ 2 is strictly increasing on [x2,x 3 ].If x3 <x 2 then Φ 2 is strictly decreasing on [x3,x 2 ].WehaveΦ 2(x α ) = R ( 1) (α) for α [0, 1]. The case x1 = x 2.Wehavexα = x1 for α [0, 1]. The case x2 = x 3.Wehavexα = x2 for α [0, 1]. Let Ψ 1 :[0, 1] [min{x 1,x 2 }, max{x 1,x 2 }], Ψ 2 :[0, 1] [min{x 2,x 3 }, max{x 2,x 3 }] be functions satisfying Ψ 1 = Φ 1 1 if x 1 = x 2 and Ψ 2 = Φ 1 2 if x 2 = x 3. The solutions xα and x α can be written as x α = Ψ 1 (L ( 1) (α)), x α = Ψ 2 (R ( 1) (α)).

10 T. Rzeżuchowski, J. W asowski / Fuzzy Sets and Systems 159 (2008) They are monotone as functions of α [0, 1] and so the next proposition follows: Proposition 4.2. Consider Eq. (20) with fuzzy parameters given by (26). Then its formal fuzzy solution ˆX can be represented as a pair of monotone functions (Ψ 1 L ( 1), Ψ 2 R ( 1) ). Moreover we have (i) ˆX is a fuzzy number iff x1 x 2 x 3 ; (ii) dual ˆX is a fuzzy number iff x3 x 2 x 1. We continue with the parameters à and B givenby(26). Whether ˆX is a fuzzy number or not depends on the relation of the support of B to the supports of remaining parameters it should be sufficiently great. We fix the fuzzy parameters à 1,...,à m, the value b in B and try to determine the range for β l and β r in the description of B for which the solution ˆX is a fuzzy number. Remark that x2 does not depend on β l and β r. Denote m m β 0 l = α l,i (x2 )i and β 0 r = α r,i (x2 )i. i=1 i=1 We see that x 1 x 2 iff β l β 0 l and x 2 x 3 iff β r β 0 r. This implies that ˆX is a fuzzy number iff β l β 0 l and β r β 0 r. In order to make the presentation simple we assumed that f in (20) is a polynomial with respect to the variable x. These considerations can be carried out with minor changes in a more general setting for the equations investigated in Section 4 in [21] Linear systems We consider in this section linear systems Ax = b, where A = (a ij ) s,n R s,n, b R s and x R n. We admit that there is an uncertainty about the parameters and it is described by a fuzzy (s n)-matrix à = (à ij ) and a fuzzy s-vector B = ( B i ). With each parameter a ij and b i of (27) is associated a quantifier or as it was described in Section 2.3. According to Definition 3.4 a fuzzy AE-solution set of type AE of this equation is described by the following set: ˆΞ AE (Ã, B) ={Ξ AE ([Ã] α, [ B] α ) 0 α 1}. It has been proved in [5] in the case when the quantifier is connected with all the parameters that under some assumptions on à and B the fuzzy AE-solution set is a regular fuzzy set. We do not know any paper where other systems of quantifiers would be investigated. In our opinion such solutions very often are not regular fuzzy sets. We give now some properties of fuzzy AE-solution sets for fuzzy parameters of the following form: à ij = (a ij, α l,ij, α r,ij ) LL, B i = (b i, β l,i, β r,i ) LL (i = 1,...,s; j = 1,...,n), (28) where L is a fixed function satisfying the conditions given in the Introduction (here R = L). Put p ij =[ α l,ij, α r,ij ], q i =[ β l,i, β r,i ] and A = (a ij ) R s,n, b = (b i ) R s, P = (p ij ) IR s,n and q = (q i ) IR s. The parameters à and B from (28) can be then represented in the following way: [Ã] α = A + L ( 1) (α) P and [ B] α = b + L ( 1) (α) q (29) (for the meaning of interval addition and multiplication in this case see for example [19]). (27)

11 2126 T. Rzeżuchowski, J. W asowski / Fuzzy Sets and Systems 159 (2008) Our analysis of fuzzy AE-solution set will be based on the following analytical characterization of interval AEsolution sets given in [19] (Theorem 3.4). Let C IR s,n and d IR s. A point x R n belongs to Ξ AE (C, d) iff the following inclusion holds: C x d d C x. To shorten the notation we put S α = Ξ AE ([Ã] α, [ B] α ).Inviewof(29) a point x R n belongs to S α iff (A x b) + L ( 1) (α) (P x q ) L ( 1) (α) (q P x). (30) Proposition 4.3. Consider the system (27) with fuzzy parameters à and B given by (28). Then (i) S 1 Ξ AE (P, q) = S 1 S β for β < 1; (ii) S β Ξ AE (P, q) S γ for γ < β. Proof. The reason for (i) is that x S 1 iff A x b = 0. To prove (ii) we put u = P x q, u = q P x and write (30) for α = γ under the form (A x b) + L ( 1) (β) u + (L ( 1) (γ) L ( 1) (β)) u L ( 1) (β) u + (L ( 1) (γ) L ( 1) (β)) u, where β > γ. This finally implies that if a point x belongs to both S β and Ξ AE (P, q) then it is also in S γ. Remark 4.2. Let S 1 =. From (i) we can see that one of the following conditions holds: (i) S 1 S α for all α [0, 1); (ii) S 1 S α = for all α [0, 1). Remark 4.3. Assume that S 0 is compact in R n.ifs α Ξ AE (P, q) for all α [0, 1] then ˆΞ AE (Ã, B) is a regular fuzzy set. Example 4.1. Consider a 2 2 linear system (27) with parameters described by the following fuzzy numbers: à 11 = à 22 = (3, 1, 1), à 12 = (0, 2, 1), à 21 = (0, 1, 2), B 1 = B 2 = (0, 2, 2). The interval AE-solution sets Ξ AE ([Ã] 0, [ B] 0 ) for various sets of quantifiers A and E can be found in [19]. Put ( ) ( ) [ 1, 1] [ 2, 1] [ 2, 2] P =, q =. [ 1, 2] [ 1, 1] [ 2, 2] It can be easily seen that here S α = Ξ tol ([Ã] α, [ B] α ) Ξ tol (P, q) for all α [0, 1] and Remark 4.3 implies that the fuzzy AE-solution set ˆΞ tol (Ã, B) ={S α 0 α 1} is a regular fuzzy set Linear systems with crisp matrices We are interested in this section in formal, fuzzy solutions of a linear system Ax = b with a crisp matrix A = (a ij ) R n n and a vector b R n described by a fuzzy n-vector B = ( B i ) composed of fuzzy numbers. We shall look for a formal solution of the form ˆX ={X α 0 α 1} where X α = (X1 α,...,xα n ) KRn and Xi α =[x α i, xα i ], [ B i ] α =[b α i, bα i ] (i = 1,...,n). According to Definition 3.3 for every α [0, 1] the interval n-vector X α should be a solution of the following system: n j=1 a ij [x α j, xα j ]=[bα i, bα i ] (i = 1,...,n) (32) (31)

12 T. Rzeżuchowski, J. W asowski / Fuzzy Sets and Systems 159 (2008) with respect to the Kaucher arithmetic. This system of interval equations is equivalent to a certain linear crisp system of 2n equations with 2n variables x α 1,...,xα n, xα 1,...,xα n : Su = v, (33) where S R 2n 2n and u, v R 2n. If the components of a solution u if (33) actually has a solution are linear combinations of b α 1,...,bα n, bα 1,...,bα n then such solution generates a family ˆX being the seeked fuzzy, formal solution. We refer now to the paper [7] where the system (31) was considered. The departure point for defining solutions apart notational differences was a solution of interval system (32) in classical interval arithmetic. Remark that the addition of intervals and multiplication of intervals by real numbers are in the classical interval arithmetic and Kaucher interval arithmetic defined by the same formulae. Having this in view we see that any solution of the crisp system (33) which satisfies the condition x α i xα i for i = 1,...,n (34) defines also a solution of (32) in the classical interval arithmetic. Remark yet that even if (34) is true for all α [0, 1] then ˆX does not necessarily belong to FI n. In the case when there is a unique solution of (32) it has been admitted in [7] that such solution always defines a fuzzy solution Û ={(U1 α,...,uα n ) (0 α 1} given by Ui α =[min{x α i, xα i, x1 i }, max{xα i, xα i, x1 i }]. (35) If ˆX FI n then Û = ˆX is called strong fuzzy solution. In the opposite case Û is called weak fuzzy solution. It should be noted that the problem of relations of weak solutions with the initial system (31) was not treated in [7]. Example 4.2. We consider the following 2 2 system as an example of (31) { x1 2x 2 = b 1, x 1 + 3x 2 = b 2, (36) where the parameters b 1 and b 2 are described by fuzzy numbers B 1 = (1, 1, 1) and B 2 = (2, 2, 2). The system (33) corresponding to (36) and B 1, B 2 has a unique solution given by x α 1 = 2.4 α, xα 1 = α + 0.4, xα 2 = α 0.8, xα 2 = 1.2 α. Observe that x α 1 <xα 1 for α [0, 1), x1 1 = x1 1 and xα 2 xα 2 for α [0, 1]. Thus we have here a weak fuzzy solution given by the intervals U α 1 =[α + 0.4, 2.4 α] and U α 2 =[α 0.8, 1.2 α], 0 α 1, (37) which constitute fuzzy numbers. Consider now the relation between the solution (37) and the initial system (36). Remark first that for α [0, 1] we have (1.4α, 0.2α) Ξ uni (A, [ B] α ) and (1.4α, 0.2α) / U1 α U 2 α. Hence for α [0, 1) the product is not a subset of the interval AE-solution set Ξ uni (A, [ B] α ). The relation itself can be described by the following statement: For any α [0, 1] x 1 U α 1, x 2 U α 2, b 1 [ B 1 ] α, b 2 [ B 2 ] α, x 1 2x 2 = b 1 x 1 + 3x 2 = b 2, (38) which means that for any x 1 U α 1 the values b 1, b 2, x 2 should be chosen according to the following recipe: (a) take b i [ B i ] α (i = 1, 2) so that the equality 3b 1 + 2b 2 = 5x 1 is satisfied; (b) put x 2 = (b 2 b 1 )/5 (which is a number within U α 2 ). Statement (38) gives a logical meaning to the weak fuzzy solution (37). In the following section we give some more remarks concerning such interpretations of formal solutions.

13 2128 T. Rzeżuchowski, J. W asowski / Fuzzy Sets and Systems 159 (2008) Some remarks concerning formal solutions We have considered in Sections 2.2 and 3.2 formal solutions of classical equations (2) with f described in (1) and parameters a, b modelled by intervals or fuzzy sets. In these considerations the function f is fixed and formal solutions are based on so-called natural interval extension (7) of f. It is well known that there are many possible interval versions of that equation. If we want to appeal to applications it is important to find the version that best fits the problem under consideration. An example of a specific problem is considered in [2] in the context of fuzzy extensions of financial mathematics. The authors propose there a different version of fuzzy Internal Rate of Return (IRR) from that given earlier by Buckley [3]. It corresponds better, in their opinion, to the reality. Our remarks will concern only the mathematical aspects of those proposals. Similarly to [2] we assume that the cash flows are modelled by positive, fuzzy intervals à i (i = 0, 1,...,n). Both in [2,3] the fuzzy IRR denoted by R is seeked by solving in classical interval arithmetic for each α-cut, respectively, the equations of the form and a α 0 = aα 1 /x + aα 2 /x2 + +a α n /xn a α 0 xn = a α 1 xn 1 + a α 2 xn 2 + +a α n, where ai α =[à i ] α and x = 1 +[ R] α is a positive interval. The following relation between the solution of these equations in the Kaucher arithmetic can be proved (using simple transformations): if x is a positive solution of any of these equations then dual x is a solution of the other. It is obvious that in the case of real parameters those equations are equivalent. In our opinion in view of interpretations (24) and (25) we cannot speak here about two essentially different definitions of fuzzy IRR. Let now ˆX ={X α 0 α 1} be a formal fuzzy solution to (2) with fuzzy parameters à and B. This means that the interval vectors X α = (X1 α,...,xα n ) are solutions to the interval equations F f ([Ã] α, x) =[ B] α (Definition 3.3). Any deeper connections of X α with (2) are interesting, especially in view of possible applications and it occurs that some descriptions of logical character can be associated with the sets X α (see [8]). We consider two cases. (1) X α IR n. Then, in view of Proposition 6.2 in [19], we have X α Ξ tol (f, [Ã] α, [ B] α ) and this inclusion provides the following logical description of X α : x X α, a [Ã] α, b [ B] α, f (a, x) = b. (2) X α / IR n. Then X α contains some improper components and it seems justifiable trying to define on that basis some real solutions U α IR n if the relations with the initial equation are explained. It should be underlined that if we have in view some real applications then such real solutions are necessary. If all the components of X α are improper then, due to Proposition 6.3 in [19], we have dual X α Ξ ctr (f, [Ã] α, [ B] α ) provided that dual F f ([Ã] α,x α ) = F f (dual [Ã] α, dual X α ) holds. This inclusion gives the following logical description where U α = dual X α x U α, b [ B] α, a [Ã] α, f (a, x) = b. When n>1 and some components of X α are proper, some other improper, the situation becomes more complicated how shows it Example 4.2. There U α = (dual X α 1,Xα 2 ) IR2 is not a subset of Ξ uni (A, [ B] α ) and we have the description (37) only the quantifier accompanies x 2. Further investigations are necessary.

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