Modeling uncertain variables of the weighted average operation by fuzzy vectors

Information Sciences 181 (2011) 4969–4992

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Information Sciences journal homepage: www.elsevier.com/locate/ins

Modeling uncertain variables of the weighted average operation by fuzzy vectors Ondrˇej Pavlacˇka ´ University Olomouc, 17. listopadu 12, Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacky 771 46 Olomouc, Czech Republic

a r t i c l e

i n f o

Article history: Received 6 April 2010 Received in revised form 13 April 2011 Accepted 3 June 2011 Available online 8 July 2011 Keywords: Fuzzy weighted average Fuzzy vector Separability of fuzzy vectors Normalized fuzzy weights Fuzzy probabilities Multiple criteria decision making

a b s t r a c t The paper deals with the fuzzy extension of the weighted average operation. First, we study the convenient ways how uncertain weights and weighted values can be modeled by fuzzy vectors. We show that, in comparison to a tuple of fuzzy numbers that have been used for modeling uncertain values of particular weights and weighted values up to now, fuzzy vectors extend the possibilities of utilizing the vague expert information concerning the weights and weighted values. Next, we focus on computation of a fuzzy weighted average of a fuzzy vector of weighted values with a fuzzy vector of weights. We derive a general formula and we study its special forms. The advantage of the approach presented in the paper is that the resulting fuzzy weighted average is not overly imprecise since every available information about its variables is involved in computation. This fact is illustrated by several examples. Finally, we briefly discuss the problem of defuzzification of the resulting fuzzy weighted average. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction A weighted average of real numbers u1, u2, . . . , um with weights w1, w2, . . . , wm is standardly defined by the following formula

aW ðw1 ; w2 ; . . . ; wm ; u1 ; u2 ; . . . ; um Þ ¼

Pm i¼1 wi  ui P ; m i¼1 wi

ð1Þ

where the weights w1, w2, . . . , wm are generally nonnegative real numbers whose sum is different from zero. If, for a specific P reason, only normalized weights are considered, i.e. if m i¼1 wi ¼ 1 is assumed, then the weighted average operation is usually given only by

aN ðw1 ; w2 ; . . . ; wm ; u1 ; u2 ; . . . ; um Þ ¼

m X

wi  ui :

ð2Þ

i¼1

Both the expressions (1) and (2) are widely applied in various mathematical models. However, the input parameters – weights and weighted values – can be uncertain. Let us discuss some typical cases where such kind of problem usually occurs. In multiple criteria decision making models, the weighted average operation is used for aggregating partial evaluations of alternatives. Weights of criteria are understood differently in different models. The most general definition says that the E-mail address: [email protected] 0020-0255/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2011.06.022

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weights of criteria are nonnegative real numbers whose ordering expresses the importance of criteria. According to this definition, the weights mean the measurements of criteria importance that are defined on an ordinal scale; the overall evaluation of an alternative is in such a case computed by (1). But in most of multiple criteria decision making models, the weights of criteria represent some kind of cardinal information about the importance of criteria. For instance, in the model of multiple criteria evaluation introduced in [28], the weights of criteria represent shares of the corresponding partial objectives of evaluation in the overall one. Such weights are normalized by their nature, and the expression (2) is applied for computing the overall evaluation. The weighted average operation is also used in discrete stochastic models of decision making under risk where the evaluations of alternatives depend on the fact which of the states of the world SoW1, SoW2, . . . , SoWm will occur. The evaluation of any alternative is a discrete random variable u that takes its values u1, u2, . . . , um with probabilities p1, p2, . . . , pm; the value pi, i 2 {1, 2, . . . , m}, expresses the probability of occurring of SoWi. The expected value of evaluation of an alternative, that is often taken into consideration when the alternative is to be compared with the others, is given by Eu = aN(p1, p2, . . . , pm; u1, u2, . . . , um). In decision making models based on the weighted average operation, we often deal with two kinds of uncertainty. First, the weighted values u1, u2, . . . , um, i.e. evaluations with respect to particular criteria in multiple criteria decision making or evaluations under possible states of the world in decision making under risk, can be uncertain. For instance, we can face incomplete or missing information about some aspect of an alternative or the expert evaluation of alternatives with respect to a qualitative criterion can be vague (see e.g. [28]). Second, the weights of criteria or the probabilities of states of the worlds are often ill-known; their values are commonly set subjectively on the basis of experts’ experiences or opinions. As both kinds of uncertainty can be sufficiently modeled by means of tools of fuzzy sets theory, it is reasonable to extend the functions aW and aN to fuzzy set arguments. The weighted average operation aN is applied also in Bayesian inference. Let X be a random variable that has m possible values, and let h be another random variable that takes the values h1, h2, . . . , hs with the so called prior probabilities p(h1), p(h2), . . . , p(hs). Let, for i 2 {1, 2, . . . , s}, p(xjhi) denote the conditional probability that X = x under the condition h = hi. If X = x0 is an observed datum, then for i = 1, 2, . . . , s, the posterior probability that h = hi is given by

pðx0 jhi Þ  pðhi Þ ; qðhi jx0 Þ ¼ Ps j¼1 pðx0 jhj Þ  pðhj Þ

i ¼ 1; 2; . . . ; s:

ð3Þ

The weighted average aN appears in (3) as the denominator; the prior probabilities play the role of normalized weights. However, prior probability distributions in Bayesian inference are often assessed subjectively by decision makers, and due to information deficiency, it is not realistic to express these assessments by exact real numbers (see [33]). In [21,22], also the conditional probabilities p(xjh1), p(xjh2), . . . , p(xjhs) were supposed to be uncertain. In fuzzy models, the most common way for an expert how to model uncertain weighted values and uncertain weights or probabilities is to describe their values separately by fuzzy numbers. The fuzzy extension of the weighted average operation to the case where the weighted values and/or the weights are modeled by fuzzy numbers has been studied since the second half of 70s. The case where only the weighted values u1, u2, . . . , um are fuzzy numbers while the weights or probabilities remain crisp were studied e.g. in [15,16,19,28]. Such fuzzy extensions of aW and aN are easy to compute since both the functions aW and aN are monotone in arguments u1, u2, . . . , um, and no external interactivity constraint is involved. Unlike this case, the fact that also the weights w1, w2, . . . , wm are considered to be fuzzy numbers makes the calculation substantially more complex. For real normalized weights, the expressions (1) and (2) coincide. However, as the functions aW and aN are given by different formulas, their extension to fuzzy weights has been studied separately. The calculation of the fuzzy extension of aW was studied e.g. in [1,3,10,11,13,17,25]; the uncertain weights are there modeled by nonnegative fuzzy numbers, and no external interactivity among the weights is considered. Increase of complexity of the calculation is caused by the fact that aW is not monotone in arguments w1, w2, . . . , wm. The calculation of the fuzzy extension of aN was studied for the first time in [4,5]. Although aN is a monotone function in all of its arguments, the calculation of its fuzzy extension is more complex because of the external interactivity constraint saying that the sum of w1, w2, . . . , wm is equal to one. This constraint implies that uncertain normalized weights or probabilities have to be modeled by means of a special structure of interactive fuzzy numbers. In the literature, such structure of fuzzy numbers is called according to its meaning – a tuple of normalized fuzzy weights [23–25,30,32], or a feasible tuple of fuzzy probabilities [22]. It represents a generalization of a tuple of reachable interval probabilities (see e.g. [2,21,34]). The extensions mentioned so far have a one aspect in common, namely that uncertain variables w1, w2, . . . , wm and u1, u2 ,. . . , um can be given only in the form of fuzzy numbers that express the possibility distributions of particular variables. However, an expert may want to add to the model also a different kind of information about the variables, e.g. their ordering, etc. A more general approach to modeling an m-tuple of uncertain normalized weights or probabilities, by means of an mdimensional fuzzy vector of normalized weights, was introduced in [23,26]. It was shown that a tuple of normalized fuzzy weights defines a fuzzy vector of normalized weights of a special kind. Besides a tuple of normalized fuzzy weights, an expertly set additional crisp relation describing the admissible combinations of the values of normalized weights was considered. Nevertheless, the weighted values were still modeled only by noninteractive fuzzy numbers.

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The aim of this paper is to study the fuzzy extensions of aW and aN to the case where both the weighted values and the weights are modeled by fuzzy vectors; the existing results concerning the fuzzy extension of aW and aN will be put into context. Special attention will be devoted to procedures of expert setting of fuzzy vectors of weights or weighted values in the models. For the purpose, we will apply the general results concerning the ways of modeling uncertain multidimensional quantities by fuzzy vectors and the computation with fuzzy vectors that were introduced in [23,27]. Nevertheless, in connection with modeling uncertain general weights, the results concerning the expert setting of fuzzy vectors that were related only to the case of a closed and convex underlying set will be extended. 2. Preliminaries 2.1. Basic notions A fuzzy set A on a nonempty set U is characterized by its membership function A : U ? [0, 1]. The family of all fuzzy sets on U will be denoted by F ðUÞ. By Core A and Supp A, we denote a core of A, Core A = {u 2 UjA(u) = 1}, and a support of A, Supp A = {u 2 UjA(u) > 0}, respectively. For any a 2 [0, 1], Aa means an a-cut of A, Aa = {u 2 UjA(u) P a}. Obviously, A0 = U. Any crisp set A # U can be viewed as a special kind of a fuzzy set on U; the characteristic function vA plays then the role of the membership function. Obviously, for a crisp set A, Aa = A holds for all a 2 (0, 1]. Each fuzzy set can be also characterized by the family of its a -cuts. In [20], it was proved that if an indexed nested family T {Aa}a2[0,1] is given such that A0 = U, Aa # Ab for all 0 6 b < a 6 1, and Aa = 06b
AðuÞ ¼

max

a2½0;1:u2Aa

a:

ð4Þ

A fuzzy set A 2 F ðUÞ is said to be a subset of B 2 F ðUÞ, written A # B, if A(u) 6 B(u) holds for all u 2 U. Obviously, A # B, if and only if Aa # Ba for any a 2 (0, 1]. The intersection of two fuzzy sets A; B 2 F ðUÞ is defined as a fuzzy set A \ B 2 F ðUÞ whose membership function is for all u 2 U given by (A \ B)(u) = min{A(u), B(u)}. Since the minimum operation is used, (A \ B)a = Aa \ Ba holds for all a 2 [0, 1]. Let U1, U2, . . . , Um be nonempty sets. The Cartesian product of fuzzy sets Ai 2 F ðU i Þ; i ¼ 1; 2; . . . ; m, is a fuzzy set A 2 F ðU 1  U 2      U m Þ; A ¼ A1  A2      Am , whose membership function is for all (u1, u2, . . . , um) 2 U1  U2      Um given by A(u1, u2, . . . , um) = min{A1(u1), A2(u2), . . . , Am(um)}. Since the minimum operation is used, Aa = A1a  A2a      Ama holds for any a 2 [0, 1]. The extension from functions having crisp arguments to functions with fuzzy set arguments is standardly done according to the extension principle. Let U and V be nonempty sets and let f : U ? V be a mapping. Then a fuzzy extension of f is the mapping fF : F ðUÞ ! F ðVÞ such that for any A 2 F ðUÞ, the membership function of fF(A) is defined for all v 2 V as follows:

fF ðAÞðv Þ ¼ For any

8 < :

sup u2U:f ðuÞ¼v

AðuÞ; if fu 2 Ujf ðuÞ ¼ v g – ;;

0;

ð5Þ otherwise:

v 2 V such that {u 2 Ujf(u) = v} – ;, the supremum in (5) can be attained, if and only if fF(A)a = f(Aa) holds for all

a 2 (0, 1], where f(Aa) = {v 2 Vjv = f(u), u 2 Aa}. For instance, this is the case where U ¼ Rm ; V ¼ Rn , f is continuous and the membership function of A is upper semicontinuous (shortly u.s.c.), i.e. if the a-cuts Aa are closed sets for all a 2 (0, 1]. 2.2. Fuzzy numbers and fuzzy vectors Fuzzy numbers were introduced for expressing uncertain values of continuous variables; they can be considered possibility distributions (see e.g. [36]). A fuzzy number is a fuzzy set X on R that fulfils the following conditions: (a) Core X – ;, (b) for all a 2 (0, 1], Xa are closed intervals, and (c) Supp X is bounded. Let Cl(A) mean the closure of the crisp set A throughout the paper. If Cl(Supp X) # Q where Q  R, then X is referred to as a fuzzy number on Q. The family of all fuzzy numbers will be denoted by F N ðRÞ, and the family of all fuzzy numbers on Q will be denoted by F N ðQ Þ. In [27], it was shown that any fuzzy number X can be also uniquely determined by a couple of functions x : ½0; 1 ! R and  x : ½0; 1 ! R that describe the minimal and maximal values of the a-cuts of X and of the closure of the support of X. The functions x and  x are left-continuous on (0, 1], right-continuous at 0 and satisfy xðaÞ 6 xðbÞ 6  xðbÞ 6  xðaÞ for all 0 6 a < b 6 1. In the sequel, the notation X ¼ f½xðaÞ;  xðaÞga2½0;1 will be used for the fuzzy number X such that X a ¼ ½xðaÞ;  xðaÞ for all a 2 (0, 1], and ClðSupp XÞ ¼ ½xð0Þ;  xð0Þ. For the membership function of the fuzzy number X ¼ f½xðaÞ;  xðaÞga2½0;1 , the following relation holds

( XðxÞ ¼

max

a2½0;1:x2½xðaÞ;xðaÞ

0;

a; for x 2 ½xð0Þ; xð0Þ; ð6Þ elsewhere:

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Let us note that if xðaÞ ¼  xðaÞ ¼ c for all a 2 [0, 1], then X represents a real number c. Furthermore, if x(a) = a and  xðaÞ ¼ b for all a 2 [0, 1], where a < b, then X represents a closed interval [a, b]. A fuzzy number X ¼ f½xðaÞ;  xðaÞga2½0;1 is called triangular if there exists a triple of real numbers x1 6 x2 6 x3 such that

xðaÞ ¼ x1 þ a  ðx2  x1 Þ and xðaÞ ¼ x3  a  ðx3  x2 Þ: 1

ð7Þ 2

3

In this paper, such a fuzzy number will be denoted by X = hx , x , x i. Since an uncertain value of one continuous variable is expressed by a fuzzy number, then, by means of analogy, a fuzzy vector is used for expressing uncertain values of an m-tuple of continuous variables (see e.g. [27]). An m-dimensional fuzzy vector X is a fuzzy set on Rm that fulfils the following conditions: (a) Core X – ;, (b) for all a 2 (0, 1], Xa are closed and convex subsets of Rm , (c) Supp X is bounded. If Cl(Supp X) # Q, where Q  Rm , then X is called a fuzzy vector on Q. The family of all m-dimensional fuzzy vectors will be denoted by F V ðRm Þ, and the family of all m-dimensional fuzzy vectors on Q  Rm will be denoted by F V ðQ Þ. Obviously, since a closed and convex subset of R is a closed interval, for m = 1, a fuzzy number is obtained, i.e. F V ðRÞ ¼ F N ðRÞ. Let us note that the membership function of any m-dimensional fuzzy vector X 2 F V ðRm Þ is u.s.c. Moreover, for all a 2 (0, 1], the a-cuts Xa are compact subsets of Rm . Let us denote the index set {1, 2, . . . , m} by Nm throughout the paper. For any i 2 Nm, the ith projection of the m-dimensional fuzzy vector X will be denoted by [X]i. It was proved in [23,27] that [X]i is a fuzzy number and its membership function is defined for all y 2 R by the following formula:

½Xi ðyÞ ¼

max

xj 2R;j2N m :xi ¼y

Xðx1 ; x2 ; . . . ; xm Þ:

ð8Þ

Hence, if an m-dimensional fuzzy vector X 2 F V ðRm Þ is considered a joint possibility distribution of an m-tuple of variables, it follows from the equality (8) that the projections [X]1, [X]2, . . . , [X]m represent the marginal possibility distributions of X. Let us recall now the concept of separability of fuzzy vectors that was introduced in [23,27]. It plays a significant role in expert’s setting of uncertain values of m-tuple of variables or in computing with fuzzy vectors. For any fuzzy vector X 2 F V ðQ Þ, where Q # Rm , the following general relationship holds

X # ð½X1  ½X2      ½Xm Þ \ Q :

ð9Þ

A fuzzy vector X 2 F V ðQ Þ is called separable on Q, if the equality holds in (9). Otherwise, a fuzzy vector X is called nonseparable on Q. If the equality in (9) holds for Q ¼ Rm , i.e. if X = [X]1  [X]2      [X]m, we simply say that a fuzzy vector X is separable. A separable fuzzy vector X 2 F V ðRm Þ is uniquely determined only by the m-tuple of its projections. In the literature (see e.g. [8,12,36]), if the joint possibility distribution is equal to the Cartesian product of its marginal possibility distributions, the fuzzy numbers representing the marginal possibility distributions are called to be noninteractive. Thus, the notion of separability of fuzzy vectors corresponds to the notion of noninteractivity of fuzzy numbers. If a fuzzy vector X is only a proper subset of the Cartesian product of its projections, we say that X is nonseparable; its projections are then called to be interactive fuzzy numbers. A nonseparable fuzzy vector X can be still separable on some crisp relation Q  Rm such that Q  [Q]1  [Q]2      [Q]m. In such a case, the nonseparability of X is caused solely by the fact that the underlying m-tuple of variables takes its values only from Q. A fuzzy vector X separable on Q is uniquely determined by the m-tuple of its projections and the crisp relation Q. Its projections, even though they are interactive, can be treated on Q as if they are noninteractive. If a crisp set Q  Rm is closed, convex and bounded, then it can be viewed as a fuzzy vector on Q of a special kind. For all i 2 Nm, [Q]i is a closed interval given by

½Q i ¼ fxi 2 Rjxi is the ith component of at least one vector x 2 Q g:

ð10Þ

It is obvious that Q = ([Q]1  [Q]2      [Q]m) \ Q, and thus, Q is a fuzzy vector separable on Q. Finally, let us note that if a relation Q  Rm can be expressed in the following way:

Q ¼ fðx; f1 ðxÞ; . . . ; fm1 ðxÞÞ 2 Rm jx 2 D # R; f i : D ! R is a uniquely invertible function for any i 2 Nm1 g;

ð11Þ

then each fuzzy vector X 2 F V ðQ Þ is separable on Q. In such a case, any fuzzy vector X 2 F V ðQ Þ is uniquely determined only by its arbitrary projection ½Xi0 ; i0 2 N m , and the relation Q as follows:

(

Xðx1 ; . . . ; xi0 ; . . . ; xm Þ ¼

½Xi0 ðxi0 Þ; for ðx1 ; . . . ; xi0 ; . . . ; xm Þ 2 Q; 0;

elsewhere:

ð12Þ

2.3. Expert setting of fuzzy vectors An expert’s opinion about the value of a one continuous uncertain variable can be sufficiently described by a fuzzy number. However, if we have an m-tuple of variables whose values are uncertain, it has not to be sufficient only to describe uncertain values of particular variables by an m-tuple of fuzzy numbers, since there can be some interactions among them. Therefore, an expert should express such an m-tuple by an m-dimensional fuzzy vector because the fuzzy vector enables to

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involve the possible interactions among the variables into the model; it plays an analogous role in fuzzy sets theory as, for instance, plays a random vector in probability theory. For an expert, direct setting of the membership function of a fuzzy vector can be in general difficult. It is more natural for him/her to express the values of particular variables separately by an m-tuple of fuzzy numbers, and to describe possible interactions among the variables. The interactions can be in most cases given by a crisp relation Q  Rm that represents the set of all admissible combinations of the values of the variables. The crisp relation Q can follow directly from the nature of the variables, e.g. normalized weights are nonnegative and their sum is equal to one, or it can reflect an expert’s opinion about the variables; such relation is usually closed and convex. This approach to setting the fuzzy vectors (Q was supposed to be closed and convex) was studied in [23,27]. Let an expert describe the uncertain values of an m-tuple of variables by fuzzy numbers X1, X2, . . . , Xm. First, let us assume that there are no interactions among the variables, i.e. they can take their values independently of each other. In such a case, the joint possibility distribution of the variables is the Cartesian product

Xsep :¼ X 1  X 2  . . .  X m :

ð13Þ

In [23,27], it was shown that Xsep is a separable m-dimensional fuzzy vector, and [Xsep]i = Xi for all i 2 Nm, i.e. the fuzzy numbers X1, X2, . . . , Xm represent the marginal possibility distributions of the variables. Now, let us assume that besides the above mentioned m-tuple of fuzzy numbers, a crisp relation Q  Rm is given that represents the set of all admissible combinations of the values of the variables. The maximal fuzzy relation (in the sense of ordering given by inclusion) expressing available information concerning the values of the variables is in this case given by

XQ :¼ ðX 1  X 2      X m Þ \ Q :

ð14Þ

In order XQ could be a fuzzy vector expressing the joint possibility distribution of the variables, the fuzzy numbers X1, X2, . . . , Xm and the relation Q have to be in some sense compatible. For a closed and convex set Q, it was proved in [23,27] that the fuzzy relation XQ is a fuzzy vector such that [XQ]i = Xi for all i 2 Nm, if and only if for all i 2 Nm and for all a 2 (0, 1], the following condition is satisfied:

For any xi 2 X ia ; there exist; for all j 2 Nm ; j – i; xj 2 X ja ; such that ðx1 ; . . . ; xi ; . . . ; xm Þ 2 Q :

ð15Þ

In such a case, XQ is a fuzzy vector separable on Q, and the fuzzy numbers X1, X2, . . . , Xm represent the marginal possibility distributions. However, it can be difficult for an expert to set the marginal possibility distributions of particular variables directly. Therefore, it was shown in [23,27] that for a closed and convex set Q, a fuzzy set XQ given by (14) is an m-dimensional fuzzy vector on Q, if and only if

ðCore X 1  Core X 2  . . .  Core X m Þ \ Q – ;:

ð16Þ

Thus, a fuzzy vector XQ expressing the joint possibility distributions of the m-tuple of variables can be given by fuzzy numbers that satisfy with respect to Q a weaker condition than (15). Such fuzzy numbers can be interpreted as estimations of the marginal possibility distributions [XQ]1, [XQ]2, . . . , [XQ]m since the following general relationships hold

½XQ i # X i

for all i 2 Nm :

ð17Þ

Nevertheless, despite the validity of the relationships (17), the fuzzy vector XQ is separable on Q, i.e. XQ = ([X]1  [X]2      [X]m) \ Q. The projections of the fuzzy vector XQ are computed as follows: For all i 2 Nm, let us denote X i ¼ f½xi ðaÞ;  xi ðaÞga2½0;1 , and ½XQ i ¼ f½xQi ðaÞ;  xQi ðaÞga2½0;1 . Then for all a 2 [0, 1] and for any i 2 Nm the following holds

xQi ðaÞ ¼ minfxi jxi P xi ðaÞ; there exist; for all j 2 Nm ; j – i; xj 2 ½xj ðaÞ; xj ðaÞ; such that ðx1 ; . . . ; xi ; . . . ; xm Þ 2 Q g; ð18Þ xQi ðaÞ ¼ maxfxi jxi 6 xi ðaÞ; there exist; for all j 2 Nm ; j – i; xj 2 ½xj ðaÞ; xj ðaÞ; such that ðx1 ; . . . ; xi ; . . . ; xm Þ 2 Q g: ð19Þ Remark 1. It is worth noting that instead of a crisp relation Q  Rm , an expert can describe the interactions among the variables by a fuzzy relation Q F 2 F ðRm Þ such that Core QF is nonempty. According to [27, Remark 16], if Q Fa are for all a 2 (0, 1] closed and convex sets (it is a characteristic property of such expertly defined fuzzy relation), then we have to replace Q by Q Fa in (15), and by Core QF in (16), respectively, in order to ensure that the fuzzy set XQ F :¼ ðX 1  X 2      X m Þ \ Q F is an m-dimensional fuzzy vector. Obviously, for computing the projections of XQ F , we can employ the formulas (18) and (19) if we replace Q by Q Fa for a 2 (0, 1], or by Cl(Supp QF) for a = 0. Later in the paper, particulary in connection with modeling uncertain general weights, we will see that it would be useful to extend these results even to the case where Q is not a closed and convex set. Therefore, we introduce here the notion of reasonability of fuzzy numbers with respect to a general set Q  Rm . We will see that if we omit the assumptions of convexity and closeness, the fuzzy numbers have to satisfy an additional condition besides (16).

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Definition 2. Let Q  Rm . An m-tuple of fuzzy numbers X1, X2, . . . , Xm is called reasonable with respect to Q, if the fuzzy set XQ given by (14) is an m-dimensional fuzzy vector on Q. Theorem 3. Let Q  Rm , and let X i 2 F N ðRÞ for all i 2 Nm. The fuzzy set XQ given by (14) is an m-dimensional fuzzy vector on Q, if and only if the fuzzy numbers X1, X2, . . . , Xm and the relation Q satisfy the condition (16), and the set QS,

Q S :¼ ðClðSupp X 1 Þ  ClðSupp X 2 Þ      ClðSupp X m ÞÞ \ Q ;

ð20Þ

is closed and convex. Proof. First, let us assume that XQ 2 F V ðQ Þ. Since Core XQ – ;, and Core XQ = (Core X1  Core X2      Core Xm) \ Q, the condition (16) is satisfied. Further, according to the assumption, Cl(Supp XQ) # Q. Hence, as Cl(Supp XQ) = (Cl(Supp X1)  Cl(Supp X2)      Cl(Supp Xm)) \ Cl(Q), it holds that Cl(Supp XQ) = QS. Thus, the set QS is closed and convex. Second, let us prove that if the set QS is closed and convex, and if the condition (16) is satisfied, then XQ 2 F V ðQ Þ. From (20), it follows that

ðClðSupp X 1 Þ  ClðSupp X 2 Þ      ClðSupp X m ÞÞ \ ðQ n Q S Þ ¼ ;:

ð21Þ

Hence, XQ = (X1  X2      Xm) \ QS. The condition (16) then implies that

ðCore X 1  Core X 2      Core X m Þ \ Q S – ;:

ð22Þ

As the set QS is closed and convex, it follows from (22) that XQ 2 F V ðQ S Þ, and since QS # Q, it also holds that XQ 2 F V ðQ Þ which completes the proof. h

In the proof of Theorem 3, it was shown that if an m-tuple of fuzzy numbers X1, X2, . . . , Xm is reasonable with respect to Q ; Q  Rm , then the fuzzy set XQ given by (14) is also a fuzzy vector on a closed and convex set QS defined by (20). Therefore, we can apply the above mentioned results regarding the case of a closed and convex set. Namely, since XQ = (X1  X2      Xm) \ QS, the fuzzy vector XQ is separable on QS, and for the projections of XQ, the relationships (17) hold. Moreover, it can be easily seen from (14) that

Q \ ððSupp X 1  Supp X 2      Supp X m Þ n Supp XQ Þ ¼ ;;

ð23Þ

which implies (see [27, Remark 9]) that the fuzzy vector XQ is also separable on Q, i.e. XQ = ([XQ]1  [XQ]2      [XQ]m) \ Q. Remark 4. For a given crisp relation Q  Rm , practical algorithms how can an expert set a corresponding reasonable m-tuple of fuzzy numbers (usually of a special type, e.g. triangular or trapezoidal) are based on some special conditions that are derived from Theorem 3. For particular crisp relations that are connected with the weighted average operation, such conditions are studied in Section 3. 2.4. Extension principle applied to fuzzy vectors In the case of the fuzzy extension of continuous real-valued functions, calculations with fuzzy vectors are based on the following well-known result, the proof of which can be found e.g. in [31]. Theorem 5. Let D # Rm ; m P 1, let f : D ! R be a continuous function, and let X 2 F V ðDÞ. Then fF(X) is a fuzzy number whose membership function is given by

( fF ðXÞðyÞ ¼

max XðxÞ; if fx 2 Djf ðxÞ ¼ yg – ;;

x2D:f ðxÞ¼y

0;

ð24Þ otherwise;

and for all a 2 (0, 1] the following holds

fF ðXÞa ¼ f ðXa Þ:

ð25Þ

ðaÞga2½0;1 , then it follows from (25) that for any a 2 (0, 1], the values y(a) and y ðaÞ Employing the notation fF ðXÞ ¼ f½yðaÞ; y can be obtained by solving the following problems of mathematical programming

yðaÞ ¼ min f ðxÞ;

ð26Þ

ðaÞ ¼ max f ðxÞ: y

ð27Þ

x2Xa

x2Xa

ðaÞ are right-continuous at 0, the values y(0) and y ð0Þ are given as follows: Furthermore, since the functions y(a) and y

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ðaÞ: ð0Þ ¼ lim y yð0Þ ¼ lim yðaÞ and y a!0þ

a!0þ

4975

ð28Þ

Complexity of the optimization problems (26) and (27) depends on the function f and on the form of a-cuts Xa. In the case of fuzzy vectors that are either separable or separable on some crisp relation Q # D, the formulas (26) and (27) can be rewritten (see [23,27]) as follows. If a fuzzy vector X is separable, i.e. X = X1  X2      Xm, where X i ¼ f½xi ðaÞ;  xi ðaÞga2½0;1 for any i 2 Nm, then for all a 2 [0, 1] the following holds

yðaÞ ¼ ðaÞ ¼ y

min

f ðx1 ; x2 ; . . . ; xm Þ;

ð29Þ

max

f ðx1 ; x2 ; . . . ; xm Þ:

ð30Þ

xi 2½xi ðaÞ;xi ðaÞ;i2N m xi 2½xi ðaÞ;xi ðaÞ;i2N m

The formulas (29) and (30) can be simplified if the function f is monotonic with respect to some variables. Let N" # Nm be an index set that contains indexes of all variables in which the function f is nondecreasing, N; # NmnN" be an index set that contains indexes of all variables in which the function f is nonincreasing but not constant, and let N⁄ = Nmn(N" [ N;). Then for all a 2 [0, 1], the following holds

yðaÞ ¼

  min f x1 ; x2 ; . . . ; xm :  2 ½xi ðaÞ; xi ðaÞ; i 2 N

xi xi xi

¼ xi ðaÞ; i 2 N" ¼ xi ðaÞ; i 2 N#

  max f xþ1 ; xþ2 ; . . . ; xþm : xþi 2 ½xi ðaÞ; xi ðaÞ; i 2 N 

ðaÞ ¼ y

ð31Þ

xþi ¼ xi ðaÞ; i 2 N" xþi

¼ xi ðaÞ; i 2 N

ð32Þ

#

If a fuzzy vector X is separable on Q # D, i.e. if X = (X1  X2      Xm) \ Q, where X i ¼ f½xi ðaÞ;  xi ðaÞga2½0;1 ; i ¼ 1; 2; . . . ; m, ðaÞ can be obtained for all form an m-tuple of fuzzy numbers that is reasonable with respect to Q, then the values y(a) and y a 2 [0, 1] as follows:

yðaÞ ¼

ðaÞ ¼ y

min

f ðx1 ; x2 ; . . . ; xm Þ;

ð33Þ

max

f ðx1 ; x2 ; . . . ; xm Þ:

ð34Þ

xi 2½xi ðaÞ;xi ðaÞ;i2N m ðx1 ;x2 ;...;xm Þ2Q

xi 2½xi ðaÞ;xi ðaÞ;i2N m ðx1 ;x2 ;...;xm Þ2Q

The formulas (33) and (34) correspond to the concept of constrained fuzzy arithmetics that was studied e.g. in [14]. Remark 6. If we have a fuzzy vector X = (X1  X2      Xm) \ QF where Q F 2 F ðDÞ is a fuzzy relation describing the ðaÞ, we have to replace in formulas (33) and (34) interactions among the variables, then for obtaining the values y(a) and y the symbol Q by Q Fa , if a 2 (0, 1], and by Cl(Supp QF), if a = 0. 2.5. Methods of defuzzification In real life applications, there is often a necessity to find a scalar representative value of an output fuzzy number. Such a process is usually called defuzzification. Many methods of defuzzification have been developed in the literature (see e.g. [6] and references therein). Some methods are based only on the membership function of a fuzzy number X ¼ f½xðaÞ;  xðaÞga2½0;1 . For instance, the center of gravity GR(X),

R þ1

x  XðxÞdx GRðXÞ :¼ 1 ; R þ1 XðxÞdx 1

ð35Þ

or the middle point of the mean interval EðXÞ,

EðXÞ :¼

Z

1

0

xðaÞ þ xðaÞ da: 2

ð36Þ

If an output fuzzy number is obtained as the result of the fuzzy extension of some real-valued function, we can apply for finding the scalar representation the concept of the central value and the expected value of a possibility distribution that was introduced in [8]. Let X 2 F V ðQ Þ, where Q # Rn , let g : Q ! R be an integrable function, and let a 2 (0, 1]. If Xa is a nondegenerated set, then the central value of g on Xa is defined in [8] by

R

CXa ðgÞ :¼

Xa

gðx1 ; x2 ; . . . ; xn Þdx1 dx2 . . . dxn R : dx1 dx2 . . . dxn Xa

ð37Þ

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Obviously, CXa ðgÞ gives the probabilistic mean value of gðU Xa Þ, where U Xa is a uniformly distributed random variable on Xa. Furthermore, if Xa is a degenerated set, then we compute CXa ðgÞ as the limit case of a uniform approximation of Xa with nondegenerated sets (see [9]). Let for e > 0,

Xa ðeÞ :¼ fy 2 Rn jthere exists x 2 Xa such that ky  xk 6 eg; then we define the central value of g on Xa as

CXa ðgÞ :¼ lim CXa ðeÞ ðgÞ: e!0þ

ð38Þ

The expected value of gF(X) can be simply defined as the mean of the central values of g on Xa over a 2 (0, 1], that is

Eðg F ðXÞÞ :¼

Z

1

CXa ðgÞda:

ð39Þ

0

Especially, if n = 1 and g  id is the identity function over R, then it can be easily seen that EðidF ðXÞÞ ¼ EðXÞ holds for any fuzzy number X. If we want to give less importance to the lower cuts of fuzzy numbers, we can employ the notion of weighting functions (see e.g. [7]). A function f : ½0; 1 ! R is called a weighting function if it is nonnegative, monotone increasing and normalized R1 over the unit interval, i.e. 0 f ðaÞda ¼ 1. Then the expected value of gF(X) with respect to a weighting function f is defined in [8] by

Ef ðg F ðXÞÞ :¼

Z

1

CXa ðgÞ  f ðaÞda:

ð40Þ

0

3. Modeling uncertain variables of the weighted average operation by fuzzy vectors In this section, we will study the convenient ways how can an expert easily describe the joint possibility distribution of an m-tuple of the weighted values u1, u2, . . . , um or of an m-tuple of the weights w1, w2, . . . , wm by an m-dimensional fuzzy vector. We will particularly focus on separability of obtained fuzzy vectors as it will play a significant role in computation of the fuzzy weighted average. 3.1. Fuzzy vectors of weighted values Generally, the weighted values u1, u2, . . . , um are not connected by any interaction. In multiple criteria decision making models, they express evaluations of an alternative according to particular criteria; they are usually nonnegative elements from a given scale. In the models of decision making under risk, they can be in particular cases also negative, e.g. under certain economic conditions representing one state of the world, we can suffer a loss if we realize a given alternative. Thus, in fuzzy models, the joint possibility distribution of the weighted values can be described by an arbitrary m-dimensional fuzzy vector U 2 F V ðRm Þ. In fuzzy extensions of both the weighted average operations (1) and (2) studied so far, the information about the weighted values was supposed to be given only by an m-tuple of noninteractive fuzzy numbers U1, U2, . . . , Um. In such a case, the fuzzy numbers represent the marginal possibility distributions of the particular weighted values, and the corresponding joint possibility distribution of the weighted values is the separable fuzzy vector Usep = U1  U2      Um. However, in the models, a different kind of information about the weighted values can be available. For instance, besides an m-tuple of fuzzy numbers U1, U2, . . . , Um that describe particular weighted values, a crisp relation Q  Rm representing the admissible combinations of the weighted values can be given. Provided that the m-tuple U1, U2, . . . , Um is reasonable with respect to Q, the corresponding joint possibility distribution of weighted values is the fuzzy vector UQ = (U1  U2      Um) \ Q that is separable on Q. Naturally, we can also consider a more complex information about the weighted values that is expressed by an m-dimensional fuzzy vector U that is not separable on any subset Q  Rm . Example 1. Let, for m = 3, the possibility distributions of particular weighted values be given by triangular fuzzy numbers U1 = h1, 3, 5i, U2 = h5, 6, 7i and U3 = h3, 6, 9i. Let us show now three examples of fuzzy vectors of weighted values that could represent the corresponding joint possibility distribution. First, let the fuzzy numbers U1, U2 and U3 be noninteractive. Then the corresponding joint possibility distribution is given by the separable fuzzy vector Usep = U1  U2  U3. Second, let all admissible combinations of weighted values be expressed by a crisp relation RU,

RU :¼ fðu1 ; u2 ; u3 Þ 2 R3 ju1 6 u2 ; u1 6 u3 g:

ð41Þ

It can be easily seen that the fuzzy numbers U1, U2 and U3 satisfy with respect to RU the condition (15). Thus, the corresponding joint possibility distribution is a fuzzy vector URU ¼ ðU 1  U 2  U 3 Þ \ RU that is separable on RU. Third, let us consider the fuzzy vector of weighted values UE 2 F V ðR3 Þ whose a-cuts are for all a 2 (0, 1] given as follows:

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Fig. 1. The 0.5-cut of the fuzzy vector of weighted values UE.

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi )  1 1  ðu1  3Þ2 þ ðu2  6Þ2 þ ðu3  6Þ2 6 1  a : ðu1 ; u2 ; u3 Þ 2 R3   4 9

( UEa ¼

ð42Þ

The 0.5-cut of UE is shown in Fig. 1. The fuzzy vector UE is neither separable nor separable on any crisp relation Q  R3 . Hence, the fuzzy numbers U1, U2 and U3 represent in this case only the projections of UE to particular dimensions. 3.2. Fuzzy vectors of normalized weights In fuzzy models, an uncertain m-tuple of normalized weights or probabilities can be described by an m-dimensional fuzzy vector W whose support Supp W is a subset of the m-dimensional probability simplex S m ,

( S m :¼

ðw1 ; w2 ; . . . ; wm Þ 2 Rm jwi P 0 for all i 2 Nm ;

m X

) wi ¼ 1 :

ð43Þ

i¼1

Such a fuzzy vector represents the joint possibility distribution of the m-tuple of normalized weights or probabilities, and its projections [W]1, [W]2, . . . , [W]m express the marginal possibility distributions of particular weights or probabilities. Since the paper deals with fuzzy extension of the weighted average operation, the fuzzy vectors on the probability simplex will be called here fuzzy vectors of normalized weights. The family of all m-dimensional fuzzy vectors of normalized weights will be denoted by F V ðS m Þ. Remark 7. The 2-dimensional probability simplex S 2 can be expressed in the following way:

S 2 ¼ fðw1 ; w2 Þ 2 R2 jw1 2 ½0; 1; w2 ¼ 1  w1 g:

ð44Þ

As the function f(x) = 1  x is uniquely invertible, S 2 satisfies the condition (11) which implies that any 2-dimensional fuzzy vector of normalized weights is separable on S 2 . This fact also means that for setting a fuzzy vector W 2 F V ðS 2 Þ, it is sufficient only to describe one weight, e.g. the first one, by a fuzzy number W 1 2 F N ð½0; 1Þ. The membership function of W is then, according to (12), given for all ðw1 ; w2 Þ 2 S 2 by W(w1, w2) = W1(w1). For m P 3, setting a fuzzy vector of normalized weights can be in general difficult. For an expert, it is more convenient to express the particular values of normalized weights or probabilities by fuzzy numbers W i 2 F N ð½0; 1Þ, i 2 Nm, and/or to describe possible additional interactions among the normalized weights or probabilities by a crisp relation Q  Rm such that Q \ S m is nonempty (see [23,26]). Let us study now this approach in more details. We will see that the corresponding fuzzy vectors of normalized weights are either separable on S m , or separable on some proper subset of S m .

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First, let us assume that an expert only describes the particular values of normalized weights or probabilities by an m-tuple of fuzzy numbers W i 2 F N ð½0; 1Þ; i ¼ 1; 2; . . . ; m. As the set of all admissible combinations of the values of normalized weights or probabilities is represented by S m , it was shown in [22,24,25,30,32] that the particular values of normalized weights or probabilities have to be expressed by a special structure of fuzzy numbers called an m-tuple of normalized fuzzy weights, or alternatively, an m-tuple of feasible fuzzy probabilities. The first terminology will be used hereafter in the paper. Definition 8. We say that fuzzy numbers W 1 ; W 2 ; . . . ; W m 2 F N ð½0; 1Þ form an m-tuple of normalized fuzzy weights if they satisfy for all a 2 (0, 1] and for all i 2 Nm the following condition:

For any wi 2 W ia ; there exist; for all j 2 Nm ; j – i; wj 2 W ja ; such that wi þ

m X

wj ¼ 1:

ð45Þ

j¼1;j–i

In such a case, the fuzzy relation expressing the whole available information about the normalized weights or probabilities is given (see [23,26]) as follows:

WSm ¼ ðW 1  W 2      W m Þ \ S m :

ð46Þ

Since the probability simplex S m is closed and convex, and since the condition (45) represents the special case of (15) for X i ¼ W i ; i ¼ 1; 2; . . . ; m; Q ¼ S m , the following holds: if the fuzzy numbers W1, W2, . . . , Wm form an m-tuple of normalized fuzzy weights, then the fuzzy relation WS m given by (46) is a fuzzy vector of normalized weights that is separable on S m . Moreover, as ½WSm i ¼ W i for all i 2 Nm, the normalized fuzzy weights represent the marginal possibility distributions of the particular weights or probabilities. The procedures of practical direct setting of an m-tuple of normalized fuzzy weights, particularly in the form of triangular or trapezoidal fuzzy numbers, or in the form of symmetric fuzzy numbers, were proposed in [24,25]. Another approach consists in estimating of normalized fuzzy weights by an m-tuple of fuzzy numbers that is reasonable with respect to S m .  i ðaÞga2½0;1 for all i 2 Nm. The m-tuple W1, W2, . . . , Wm is reasonable with respect Theorem 9. Let W i 2 F N ð½0; 1Þ; W i ¼ f½wi ðaÞ; w to S m , if and only if the following holds m X i¼1

wi ð1Þ 6 1 6

m X

 i ð1Þ: w

ð47Þ

i¼1

Proof. Since S m is closed and convex, it is sufficient only to show that

ðCore W 1  Core W 2      Core W m Þ \ S m – ;:

ð48Þ

 i ð1Þ for all i 2 Nm, it is obvious that the conditions (47) and (48) are equivalent. h As Core W i ¼ ½wi ð1Þ; w For an expert, it is obviously much easier to describe the particular values of normalized weights or probabilities by fuzzy numbers that satisfy only the condition (47) instead of (45). For instance, it is sufficient for him/her only to set three m-tuP Pm 2 Pm 3 1 ples of values from the unit interval wj1 ; wj2 ; . . . ; wjm , j = 1, 2, 3, such that m i¼1 wi 6 1; i¼1 wi ¼ 1, and i¼1 wi P 1, in order to define an m-tuple of triangular fuzzy numbers hw1i ; w2i ; w3i i; i ¼ 1; 2; . . . ; m, reasonable with respect to S m .  i ðaÞga2½0;1 , i = 1, 2, . . ., m, form a reasonable m-tuple with respect to S m , then the fuzzy If fuzzy numbers W i ¼ f½wi ðaÞ; w relation WS m given by (46) that expresses the whole available information about the normalized weights or probabilities is a fuzzy vector of normalized weights separable on S m , and ½WSm i # W i for all i 2 Nm. According to the formulas (18)  S m i ðaÞga2½0;1 , i = 1, 2, . . . , m, are given as follows: and (19), the projections ½WS m i ¼ f½wSm i ðaÞ; w

(

wSm i ðaÞ ¼ max wi ðaÞ; 1  (  Sm i ðaÞ ¼ min w  i ðaÞ; 1  w

m X j¼1;j–i m X

)

 j ðaÞ ; w

ð49Þ

) wj ðaÞ :

ð50Þ

j¼1;j–i

As the projections ½WSm 1 ; ½WS m 2 ; . . . ; ½WSm m express the marginal possibility distributions of particular weights or probabilities, they form an m-tuple of normalized fuzzy weights estimated by fuzzy numbers W1, W2, . . . , Wm. The formulas (49) and (50), that are here viewed as special forms of (18) and (19), were introduced in [23,32]. However, in [32], they were interpreted as normalization of dependent fuzzy weights. Only the special forms of (49) and (50) where the fuzzy numbers W1, W2, . . . , Wm are of a triangular or trapezoidal shape were derived in [22,25,30]. Example 2. Let, for m = 3, an expert describe the uncertain values of normalized weights only by the following triple of triangular normalized fuzzy weights (see Fig. 2):

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Fig. 2. Normalized fuzzy weights W1, W2 and W3.

W 1 ¼ h0; 0:35; 0:5i ¼ f½0:35a; 0:5  0:15aga2½0;1 ;

ð51Þ

W 2 ¼ h1=3; 0:4; 1i ¼ f½1=3 þ a=15; 1  0:6aga2½0;1 ;

ð52Þ

W 3 ¼ h0; 0:25; 1=3i ¼ f½0:25a; 1=3  a=12ga2½0;1 :

ð53Þ

The corresponding fuzzy vector of normalized weights is then the fuzzy set WS3 ¼ ðW 1  W 2  W 3 Þ \ S 3 . For all a 2 (0, 1], the a-cuts WS3 a are given as follows:

WS3 a ¼ fðw1 ; w2 ; w3 Þ 2 S 3 jw1 2 ½0:35a; 0:5  0:15a; w2 2 ½1=3 þ a=15; 1  0:6a; w3 2 ½0:25a; 1=3  a=12g:

ð54Þ

The membership function of the fuzzy vector of normalized weights WS 3 is depicted on the underlying 3-dimensional probability simplex S 3 in Fig. 3. Another possibility of setting is that an expert gives only the information about additional interactions among normalized weights or probabilities by a crisp relation Q  Rm such that T ¼ Q \ S m is a nonempty, closed, and convex set (t is always bounded because S m is bounded). In such a case, the relation T represents a special kind of a fuzzy vector of normalized weights such that Ta = T for all a 2 (0, 1] (see [23,26]). For all i 2 Nm, the marginal possibility distributions are closed intervals given by (10). It can be easily seen that T is separable on itself. Example 3. Let, for m = 3, an expert give only the ordinal information about the normalized weights by a crisp relation RW,

RW :¼ fðw1 ; w2 ; w3 Þ 2 S 3 jw3 6 w1 6 w2 g:

ð55Þ

Since RW is a crisp, closed, convex and bounded set, it represents a special kind of 3-dimensional fuzzy vector of normalized weights that is separable on itself. According to (10), the marginal possibility distributions are closed intervals [RW]1 = [0, 0.5], [RW]2 = [1/3, 1] and [RW]3 = [0, 1/3]. Naturally, an expert can also give both information about the particular values of normalized weights or probabilities by fuzzy numbers W 1 ; W 2 ; . . . ; W m 2 F N ð½0; 1Þ, and information about additional interactions among the normalized weights or probabilities by a crisp relation Q  Rm such that the set T ¼ Q \ S m is nonempty. In this case, the fuzzy relation expressing the whole available information about the normalized weights or probabilities is given (see [23,26]) as follows:

WT ¼ ðW 1  W 2      W m Þ \ T:

ð56Þ

If the m-tuple of fuzzy numbers W1, W2 ,. . . , Wm is reasonable with respect to T, then WT is a fuzzy vector of normalized weights separable on T. The marginal possibility distributions of particular weights or probabilities are the projections [WT]1, [WT]2, . . . , [WT]m. They can be obtained from the fuzzy numbers W1, W2, . . . , Wm according to the formulas (18) and (19). Example 4. Let, for m = 3, an expert set both the ordinal information about the normalized weights by a crisp relation RW given by (55) and the cardinal information about the values of weights by the triple of triangular normalized fuzzy weights W1, W2 and W3 given by (51)–(53). Since the fuzzy numbers W1,W2 and W3 satisfy with respect to RW the condition (15), the fuzzy set WRW ¼ ðW 1  W 2  W 3 Þ \ RW is a fuzzy vector of normalized weights separable on RW whose projections are identical with the fuzzy numbers W1, W2 and W3. The a-cuts of WRW are for all a 2 (0, 1] given as follows:

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Fig. 3. The membership function of the fuzzy vector of normalized weights WS3 ¼ ðW 1  W 2  W 3 Þ \ S 3 .

WRW a ¼ fðw1 ; w2 ; w3 Þ 2 S 3 jw3 6 w1 6 w2 ; w1 2 ½0:35a; 0:5  0:15a; w2 2 ½1=3 þ a=15; 1  0:6a; w3 2 ½0:25a; 1=3  a=12g:

ð57Þ

The membership function of the fuzzy vector of normalized weights WRW is depicted on the underlying 3-dimensional probability simplex S 3 in Fig. 4. It can be easily seen that although the normalized fuzzy weights W1, W2 and W3 satisfy W3 6 W1 6 W2 (in the sense of ordering given by a-cuts), the fuzzy vector of normalized weights WRW is not separable on S 3 . It follows from the fact that for some a 2 (0, 1], the a-cuts of W1,W2 and W3 are not disjoint. Thus, the crisp relation RW really represents an additional condition for the model. Naturally, we can consider also a fuzzy vector of normalized weights W that is not separable on any crisp relation Q # S m . For instance, such a fuzzy vector can be obtained as the result of the process of normalization of a fuzzy vector of general weights (see [23]). 3.3. Fuzzy vectors of general weights In some models, it is not assumed that the sum of the weights is strictly equal to one. The weights are only supposed to be m nonnegative real numbers w1, w2, . . . , wm whose sum is different from zero. If we denote W m :¼ ðRþ 0 Þ n fog, where o denotes the zero vector of proper dimension, then the joint possibility distribution of an m-tuple of such weights can be described by an m-dimensional fuzzy vector W 2 F V ðW m Þ. As well as in the case of weighted values, in fuzzy extensions of the weighted average operation (1) studied so far, the information about the weights was supposed to be given only by an m-tuple of fuzzy numbers W i 2 F N ðRþ 0 Þ, i = 1, 2, . . . , m.

Fig. 4. The membership function of the fuzzy vector of normalized weights WRW ¼ ðW 1  W 2  W 3 Þ \ RW .

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Since W m represents the underlying set of all admissible vectors of weights, the m-tuple W1,W2,. . .,Wm has to be reasonable with respect to W m . Practical setting of such fuzzy numbers is based on the following theorem.  Theorem 10. Let W i 2 F N ðRþ 0 Þ; W i ¼ f½wi ðaÞ; wi ðaÞga2½0;1 for all i 2 Nm. The m-tuple W1, W2, . . . , Wm is reasonable with respect to W m , if and only if

wi ð0Þ > 0 for at least one i 2 Nm :

ð58Þ

 1 ð0Þ  ½w2 ð0Þ; w  2 ð0Þ      ½wm ð0Þ; w  m ð0Þ. As W m is convex but not closed, we have to Proof. Let us denote W 0 :¼ ½w1 ð0Þ; w show according to Theorem 3 that the condition (58) is satisfied, if and only if

 1 ð1Þ  ½w2 ð1Þ; w  2 ð1Þ      ½wm ð1Þ; w  m ð1ÞÞ \ W m – ;; ð½w1 ð1Þ; w

ð59Þ

and the set W mS :¼ W 0 \ W m is closed and convex.  j ð0Þ  Rþ First, let us assume that the condition (58) is satisfied. Then o R W0, and since ½wj ð0Þ; w 0 for all j 2 Nm, it holds that W 0  W m . Hence, W mS ¼ W 0 , and thus, W mS is closed and convex. The inequality (59) follows from the fact that  j ð1Þ 6 w  j ð0Þ holds for all j 2 Nm. wj ð0Þ 6 wj ð1Þ 6 w On the contrary, let W mS be closed and convex, and let the inequality (59) hold. From (59), it follows that W mS is nonempty. Since W mS is closed, o R W0, which implies the validity of the condition (58). h If the weights are described only by fuzzy numbers W1, W2, . . . , Wm that form an m-tuple reasonable with respect to W m , then the joint possibility distribution of the weights is the fuzzy vector WW m 2 F V ðW m Þ,

WW m ¼ ðW 1  W 2      W m Þ \ W m ;

ð60Þ

that is separable on W m . Since W 1  W 2      W m  W m , it is obvious that WW m ¼ W 1  W 2      W m , i.e. that the fuzzy vector WW m is separable. Hence, the fuzzy numbers W1,W2,. . .,Wm represent the marginal possibility distributions of the particular weights and are noninteractive. However, similarly like in the case of the weighted values, we can also consider a more complex information about the weights. Together with an m-tuple of fuzzy numbers that model the values of particular weights, an expert can describe possible interactions among the weights by a crisp relation Q  Rm such that the set T ¼ Q \ W m is nonempty. Naturally, the mtuple of fuzzy numbers has to be in such a case reasonable with respect to T. The corresponding fuzzy vector that represents the joint possibility distribution of the weights is then separable on T. Another possibility is that an expert describes the weights by a fuzzy vector W 2 F V ðW m Þ that is not separable on any crisp relation Q  W m . Finally, let us note that analogously like in the crisp case, every fuzzy vector of normalized weights obviously represents a special kind of a fuzzy vector on W m . Example 5. Let, for m = 3, an expert describe uncertain values of general weights by a triple of triangular fuzzy numbers

W 1 ¼ h0; 2; 4i ¼ f½2a; 4  2aga2½0;1 ;

ð61Þ

W 2 ¼ h0; 3; 6i ¼ f½3a; 6  3aga2½0;1 ;

ð62Þ

W 3 ¼ h0; 4; 8i ¼ f½4a; 8  4aga2½0;1 ;

ð63Þ

and say that ‘‘the sum of the first two weights is greater by one than the third weight’’. This means that the set of all admissible values of weights is expressed by a crisp relation T,

T :¼ fðw1 ; w2 ; w3 Þ 2 W 3 jw1 þ w2 ¼ w3 þ 1g:

ð64Þ

Since the fuzzy numbers W1, W2 and W3 obviously satisfy the condition (15) with respect to T, they represent the marginal possibility distributions of weights, and the joint possibility distribution of weights is expressed by a fuzzy vector WT = (W1  W2  W3) \ T that is separable on T. The a-cuts of WT are for all a 2 (0, 1] given as follows:

WT a ¼ fðw1 ; w2 ; w3 Þ 2 W 3 jw1 þ w2 ¼ w3 þ 1; w1 2 ½2a; 4  2a; w2 2 ½3a; 6  3a; w3 2 ½4a; 8  4ag:

ð65Þ

At the end of this example, let us note that since the fuzzy numbers W1, W2 and W3 do not satisfy the condition (58), they are not reasonable with respect to W 3 , and therefore, they cannot be noninteractive; they can represent the marginal possibility distributions of weights only if some additional information about the weights is given. However, in the next section, we will discuss how can we approximate the case when such an m-tuple of fuzzy numbers not reasonable with respect to W m represents the only information about the weights. 4. Fuzzy extension of the weighted average operation In this section, the fuzzy extensions of the weighted average operation aW as well as of its special form aN will be introduced. Particularly, we will focus on the case where both the weights and the weighted values are modeled by fuzzy vectors; the methods of calculation for the special kinds of input fuzzy vectors will be studied.

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The weighted average operation aW given by (1) is a continuous real function defined on W m  Rm , and thus, for all W W W X 2 F V ðW m  Rm Þ; aW F ðXÞ is a fuzzy number. If we denote aF ðXÞ ¼ f½aX ðaÞ; aX ðaÞga2½0;1 , then according to (26) and (27), for all a 2 (0, 1], the following holds

Pm i¼1 wi  ui P ; m ðw1 ;w2 ;...;wm ;u1 ;u2 ;...;um Þ2Xa wi Pm i¼1 i¼1 wi  ui X ðaÞ ¼ P a max : m ðw1 ;w2 ;...;wm ;u1 ;u2 ;...;um Þ2Xa i¼1 wi

aW X ðaÞ ¼

min

ð66Þ ð67Þ

W The values aW X ð0Þ and aX ð0Þ are obtained according to (28) as follows: W W W aW and a X ð0Þ ¼ lim aX ðaÞ X ð0Þ ¼ lim aX ðaÞ:

a!0þ

a!0þ

ð68Þ

Similarly, the weighted average aN given by (2), that in fact represents the restriction of aW to S m  Rm , is a continuous real NX ðaÞga2½0;1 , where function defined on S m  Rm . Hence, for all X 2 F V ðS m  Rm Þ; aNF ðXÞ ¼ f½aNX ðaÞ; a

aNX ðaÞ ¼ NX ðaÞ ¼ a

m X

min

ðw1 ;w2 ;...;wm ;u1 ;u2 ;...;um Þ2Xa

max

ðw1 ;w2 ;...;wm ;u1 ;u2 ;...;um Þ2Xa

i¼1 m X

wi  ui ;

ð69Þ

wi  ui

ð70Þ

i¼1

for all a 2 (0, 1], and

NX ð0Þ ¼ lim a NX ðaÞ: aNX ð0Þ ¼ lim aNX ðaÞ and a a!0þ

a!0þ

ð71Þ

As it was already mentioned in the paper, the input variables of aW and aN, i.e. the weights and the weighted values, are in decision making models often ill-known. In previous section, it was shown that their uncertain values can be adequately modeled by fuzzy vectors. Since the weights are in the models usually supposed to be independent of the weighted values, only the fuzzy sets X = W  U, where W 2 F V ðW m Þ is a fuzzy vector of weights, or W 2 F V ðS m Þ is a fuzzy vector of normalized weights, respectively, and U 2 F V ðRm Þ is a fuzzy vector of weighted values, will be considered further in this paper. The following theorem shows that the fuzzy set W  U is a fuzzy vector on W m  Rm , or a fuzzy vector on S m  Rm , respectively. Theorem 11. Let Q i # Rni for all i 2 Nm, and let Q = Q1  Q2      Qm. Let Xi 2 F V ðQ i Þ for all i 2 Nm. Then the Cartesian product P X = X1  X2      Xm is an n-dimensional fuzzy vector on Q, where n ¼ m i¼1 ni . Proof. Since Core Xi – ; for all i 2 Nm, Core X = Core X1  Core X2      Core Xm – ;. As Xia are closed and convex sets for all

a 2 (0, 1] and all i 2 Nm, Xa = X1a  X2a      Xma is a closed and convex set. Finally, boundedness of the supports Supp Xi, i = 1, 2, . . . , m, implies that Supp X = Supp X1  Supp X2      Supp Xm is a bounded set too. As Cl(Supp Xi) # Qi for all i 2 Nm, Cl(Supp X) # Q, which completes the proof. h W given by (1). Let W 2 F V ðW m Þ, and Definition 12. Let aW F be the fuzzy extension of the weighted average operation a m W U 2 F V ðR Þ. The fuzzy number U ¼ aF ðW  UÞ is called a fuzzy weighted average of a fuzzy vector of weighted values U with a fuzzy vector of weights W.

The general formulas (66) and (67) can be in the case of a fuzzy weighted average of a fuzzy vector of weighted values U W W with a fuzzy vector of weights W rewritten as follows. Let us denote aW F ðW  UÞ ¼ f½aWU ðaÞ; aWU ðaÞga2½0;1 . Then for all a 2 (0, 1], the following holds

aW WU ðaÞ ¼

W a WU ðaÞ ¼

min

Pm i¼1 wi  ui P ; m i¼1 wi

ð72Þ

max

Pm i¼1 wi  ui P : m i¼1 wi

ð73Þ

ðw1 ;w2 ;...;wm Þ2Wa ðu1 ;u2 ;...;um Þ2Ua

ðw1 ;w2 ;...;wm Þ2Wa ðu1 ;u2 ;...;um Þ2Ua

W The values aW WU ð0Þ and aWU ð0Þ are according to (68), given by W W W aW and a WU ð0Þ ¼ lim aWU ðaÞ WU ð0Þ ¼ lim aWU ðaÞ:

a!0þ

a!0þ

ð74Þ

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N Definition 13. Let aN F be the fuzzy extension of the weighted average operation a given by (2). Let W 2 F V ðS m Þ, and U 2 F V ðRm Þ. The fuzzy number U ¼ aN ðW  UÞ is called a fuzzy weighted average of a fuzzy vector of weighted values U with a F fuzzy vector of normalized weights W. NWU ðaÞga2½0;1 , then according to (69) and (70), for all a 2 (0, 1], the following holds If we denote aNF ðW  UÞ ¼ f½aNWU ðaÞ; a

aNWU ðaÞ ¼

NWU ðaÞ ¼ a

min

m X

ðw1 ;w2 ;...;wm Þ2Wa ðu1 ;u2 ;...;um Þ2Ua i¼1

max

m X

ðw1 ;w2 ;...;wm Þ2Wa ðu1 ;u2 ;...;um Þ2Ua i¼1

wi  ui ;

ð75Þ

wi  ui ;

ð76Þ

and according to (71), it holds that

NWU ðaÞ: NWU ð0Þ ¼ lim a aNWU ð0Þ ¼ lim aNWU ðaÞ and a a!0þ

a!0þ

ð77Þ

The following theorem shows that for fuzzy vectors of normalized weights, analogously as in the crisp case, the fuzzy N extensions aW F and aF coincide. Thus, we can study further only the formulas (72) and (73), since they in the case of a fuzzy vector of normalized weights embody the formulas (75) and (76). W N N Theorem 14. Let aW F and aF be the fuzzy extensions of a given by (1), and a given by (2), respectively. Then for all W 2 F V ðS m Þ and for all U 2 F V ðRm Þ, the following equality holds

N aW F ðW  UÞ ¼ aF ðW  UÞ:

ð78Þ

N W N Proof. It can be easily seen from (72), (73), (75) and (76) that if Supp W # S m , then aW WU ðaÞ ¼ aWU ðaÞ and aWU ðaÞ ¼ aWU ðaÞ hold for all a 2 (0, 1]. h

Complexity of the optimization problems (72) and (73) depends on the form of a-cuts of the input fuzzy vectors W and U. In the previous section, we have shown that the fuzzy vectors of weights or weighted values are in the models mostly separable or separable on some crisp relation. In such cases, the formulas (72) and (73) can be rewritten according to (29), (30) or (33), (34), respectively. For instance, if UQ is a fuzzy vector of weighted values separable on some crisp relation Q  Rm , i.e.  i ðaÞga2½0;1 ; i ¼ 1; 2; . . . ; m, then the fuzzy weighted average of the fuzzy UQ ¼ ðU 1  U 2  . . .  U m Þ \ Q , where U i ¼ f½ui ðaÞ; u W W vector of weighted values UQ with a fuzzy vector of weights W is a fuzzy number aW F ðW  UQ Þ ¼ f½aW;UQ ðaÞ; aW;UQ ðaÞga2½0;1 , W  where the values aW ð a Þ and a ð a Þ are obtained for all a 2 (0, 1] as follows: W;UQ W;UQ

aW W;UQ ðaÞ ¼

min ðw1 ; w2 ; . . . ; wm Þ 2 Wa  i ðaÞ; i 2 Nm ui 2 ½ui ðaÞ; u

Pm i¼1 wi  ui P ; m i¼1 wi

ð79Þ

ðu1 ; u2 ; . . . ; um Þ 2 Q W a W;UQ ðaÞ ¼

max ðw1 ; w2 ; . . . ; wm Þ 2 Wa  i ðaÞ; i 2 Nm ui 2 ½ui ðaÞ; u

Pm i¼1 wi  ui P : m i¼1 wi

ð80Þ

ðu1 ; u2 ; . . . ; um Þ 2 Q Moreover, since the weighted average operation aW is an increasing function in variables u1, u2, . . . , um, the general optimization problems (72) and (73) can be substantially simplified, if a fuzzy vector of weighted values is separable. W Definition 15. Let aW given by (1). Let W 2 F V ðW m Þ, and F be the fuzzy extension of the weighted average operation a Usep 2 F V ðRm Þ; Usep ¼ U 1  U 2  . . .  U m , where U i 2 F N ðRÞ for all i 2 Nm. The fuzzy number U ¼ aW F ðW  Usep Þ is called a fuzzy weighted average of fuzzy numbers U1, U2, . . . , Um with a fuzzy vector of weights W. W W  i ðaÞga2½0;1 for all i 2 Nm, and aW Let us denote U i ¼ f½ui ðaÞ; u F ðW  Usep Þ ¼ f½aW;Usep ðaÞ; aW;Usep ðaÞga2½0;1 . According to (31) and (32), for all a 2 (0, 1], the following holds

aW W;Usep ðaÞ ¼ W a W;Usep ðaÞ ¼

min

ðw1 ;w2 ;...;wm Þ2Wa

max

ðw1 ;w2 ;...;wm Þ2Wa

Pm i¼1 wi  ui ðaÞ Pm ; i¼1 wi Pm  i¼1 wi  ui ðaÞ Pm : w i¼1 i

ð81Þ ð82Þ

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4984

Thus, complexity of the calculation of a fuzzy weighted average of fuzzy numbers depends in this case only on the form of the a-cuts of a fuzzy vector of weights W. Remark 16. In general, the formulas (81) and (82) can be used for computing the fuzzy weighted average of a fuzzy vector of  i ðaÞga2½0;1 for all i 2 Nm, then weighted values U 2 F V ðRm Þ that satisfies the following condition: if we denote ½Ui ¼ f½ui ðaÞ; u

ðu1 ðaÞ; u2 ðaÞ; . . . ; um ðaÞÞ 2 Ua

 1 ðaÞ; u  2 ðaÞ; . . . ; u  m ðaÞÞ 2 Ua and ðu

ð83Þ

hold for all a 2 (0, 1]. An example of a fuzzy vector that satisfies (83) but is not separable is the fuzzy vector URU from Example 1. Finally, let us focus on two special cases that have been widely studied in the literature. In addition to the separable fuzzy vector of weighted values Usep, let the fuzzy vector of weights be either separable, or separable on S m . In both cases, the optimization problems (81) and (82) can be further simplified. For fuzzy numbers modeling the particular weights, we will use  i ðaÞga2½0;1 , i = 1, 2, . . . , m. the notation W i ¼ f½wi ðaÞ; w Definition 17. Let aW be the fuzzy extension of the weighted average operation aW given by (1). Let F WW m 2 F V ðW m Þ; WW m ¼ W 1  W 2      W m , where W1, W2 , . . . , Wm are fuzzy numbers on Rþ 0 satisfying (58), and let Usep 2 F V ðRm Þ, Usep = U1  U2      Um, where U i 2 F N ðRÞ for all i 2 Nm. The fuzzy number U ¼ aW F ðWW m  Usep Þ is called a fuzzy weighted average of fuzzy numbers U1, U2, . . . , Um with fuzzy weights W1, W2, . . . , Wm. W W If we denote aW F ðWW m  Usep Þ ¼ f½aWW m ;Usep ðaÞ; aWW m ;Usep ðaÞga2½0;1 , then the optimization problems (81) and (82) can be rewritten as follows:

aW WW m ;Usep ðaÞ ¼

min

 i ðaÞ;i2N m wi 2½wi ðaÞ;w

W a WW m ;Usep ðaÞ ¼

max

 i ðaÞ;i2N m wi 2½wi ðaÞ;w

Pm i¼1 wi  ui ðaÞ Pm ; wi Pm i¼1  i¼1 wi  ui ðaÞ Pm : w i¼1 i

ð84Þ ð85Þ

In [11], it was shown that using the Charnes and Coooper’s linear transformation, the nonlinear fractional programming problems (84) and (85) can be transformed to the following linear programming problems

aW WW m ;Usep ðaÞ ¼

vi

min  i ðaÞ; i 2 Nm 2 ½z  wi ðaÞ; z  w Pm v ¼ 1 i¼1 i

m X

v i  ui ðaÞ;

i¼1

ð86Þ

zP0 W a WW m ;Usep ð

aÞ ¼

vi

max  i ðaÞ; i 2 Nm 2 ½z  wi ðaÞ; z  w Pm v ¼ 1 i i¼1

m X i¼1

v i  ui ðaÞ: ð87Þ

zP0 Definition 18. Let aN be the fuzzy extension of the weighted average operation aN given by (2). Let F WSm 2 F V ðS m Þ; WSm ¼ ðW 1  W 2      W m Þ \ S m , where W1, W2, . . . , Wm form an m-tuple of normalized fuzzy weights, and let Usep 2 F V ðRm Þ; Usep ¼ U 1  U 2      U m , where U i 2 F N ðRÞ for all i 2 Nm. The fuzzy number U ¼ aN F ðWS m  Usep Þ is called a fuzzy weighted average of fuzzy numbers U1, U2, . . . , Um with normalized fuzzy weights W1, W2, . . . , Wm. NW ;Usep ðaÞga2½0;1 , then the optimization problems (81) and (82) can be exIf we denote aNF ðWS m  Usep Þ ¼ f½aNWSm ;Usep ðaÞ; a Sm pressed as follows:

aNWS

m ;Usep

ðaÞ ¼

min wi 2½wi ðaÞ;w Pm  i ðaÞ;i2Nm i¼1

NW ;Usep ðaÞ ¼ a Sm

wi ¼1

max wi 2½wi ðaÞ;w Pm  i ðaÞ;i2Nm i¼1

wi ¼1

m X

wi  ui ðaÞ;

ð88Þ

 i ðaÞ: wi  u

ð89Þ

i¼1

m X i¼1

However, it is not necessary to solve the linear programming problems (88) and (89). The following effective algorithm NW ;Usep ðaÞ was introduced in [24]. It represents a generalization of an algorithm for for computing the values aNWSm ;Usep ðaÞ and a Sm computing the expected value of a discrete random variable with interval probabilities that was introduced in [2]. m For each a 2 [0, 1], let fik gk¼1 be such a permutation on an index set Nm that ui1 ðaÞ 6 ui2 ðaÞ 6    6 uim ðaÞ. For k 2 Nm, let us denote

O. Pavlacˇka / Information Sciences 181 (2011) 4969–4992

wik ðaÞ ¼ 1 

k1 X

 ij ðaÞ  w

j¼1

m X

wij ðaÞ:

4985

ð90Þ

j¼kþ1

 ik ðaÞ. Then Let k⁄ 2 Nm be such an index that wik ðaÞ 6 wik ðaÞ 6 w  kX 1

aNWSm ;Usep ðaÞ ¼

m X

 ij ðaÞ  uij ðaÞ þ wik ðaÞ  uik ðaÞ þ w

wij ðaÞ  uij ðaÞ:

ð91Þ

j¼k þ1

j¼1

   Let fih gm h¼1 be such a permutation on an index set Nm that ui1 ðaÞ P ui2 ðaÞ P    P uim ðaÞ. For h 2 Nm, let us denote

wih ðaÞ ¼ 1 

h1 X

 ij ðaÞ  w

j¼1

m X

wij ðaÞ:

ð92Þ

j¼hþ1

 ih ðaÞ. Then Let h⁄ 2 Nm be such an index that wih ðaÞ 6 wih ðaÞ 6 w

NW a S

m

;Usep ðaÞ ¼

 hX 1

 ij ðaÞ þ wih ðaÞ  u  ih ðaÞ þ  ij ðaÞ  u w

m X

 ij ðaÞ: wij ðaÞ  u

ð93Þ

j¼h þ1

j¼1

Example 6. Let us consider the fuzzy vectors of weighted values Usep and UE from Example 1, the fuzzy vectors of normalized weights WS3 from Example 2 and WRW from Example 4, and the fuzzy vector of weights Wt from Example 5. In this example, we will focus on computation of the following fuzzy weighted averages

U N :¼ aNF ðWS3  Usep Þ ¼ f½aNWS

3

U R :¼ aNF ðWRW  Usep Þ ¼ f½aNWR

;Usep ð

W

U EN :¼

aNF ðWS3

U ER :¼

aNF ðWRW

U T :¼

aW F ðWT

U ET :¼

;Usep ð

 UE Þ ¼

f½aNWS ;UE ð 3

 UE Þ ¼

f½aNWR ;UE ð

 Usep Þ ¼

f½aW WT ;Usep ð

aW F ðWT

 UE Þ ¼

aÞ; aNWS3 ;Usep ðaÞga2½0;1 ; W

a

aÞga2½0;1 ;

NW ;U ð Þ; a R E

a

a

aÞga2½0;1 ;

;Usep ð

NW ;U ð Þ; a S3 E

W

f½aW WT ;UE ð

aÞ; aNWR

W

W Þ; a Wt ;Usep ð

W Þ; a Wt ;UE ð

a

aÞga2½0;1 ;

aÞga2½0;1 ;

aÞga2½0;1 :

We will also briefly discuss the defuzzification of these fuzzy numbers. In order to simplify the notation, we will denote the fuzzy numbers U1, U2, and U3 from Example 1 modeling the particular weighted values by

 i ðaÞga2½0;1 ; U i ¼ f½ui ðaÞ; u

i ¼ 1; 2; 3;

the triple of normalized fuzzy weights W1, W2, and W3 from Example 2 by

 Ni ðaÞga2½0;1 ; W i ¼ f½wNi ðaÞ; w

i ¼ 1; 2; 3;

and the triple of fuzzy weights W1, W2, and W3 from Example 5 by

 i ðaÞga2½0;1 ; W i ¼ f½wi ðaÞ; w

i ¼ 1; 2; 3:

Since the fuzzy vector of weighted values Usep is separable, and the fuzzy vector of normalized weights WS3 is separable on S 3 , the fuzzy weighted average UN can be calculated by the algorithms (91) and (93). In the case of the fuzzy weighted average UR, we can take advantage of both the separability of Usep and of the fact that N WRW is separable on RW. For all a 2 (0, 1], the values of aN WRW ;Usep ðaÞ and aWRW ;Usep ðaÞ are obtained by solving the following linear programming problems

aNWR

W

;Usep ð

aÞ ¼

w1  u1 ðaÞ þ w2  u2 ðaÞ þ w3  u3 ðaÞ; min  Ni ðaÞ; i 2 N3 wi 2 ½wNi ðaÞ; w P3 i¼1 wi ¼ 1

ð94Þ

w2  w1 P 0 w1  w3 P 0 NW a R

W

;Usep ð

aÞ ¼

 1 ðaÞ þ w2  u  2 ðaÞ þ w3  u  3 ðaÞ: w1  u max  Ni ðaÞ; i 2 N3 wi 2 ½wNi ðaÞ; w P3 i¼1 wi ¼ 1 w2  w1 P 0 w1  w3 P 0

ð95Þ

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The membership functions of the fuzzy numbers UN and UR are shown in Fig. 5. We can see that the fuzzy number UR is less uncertain. This means that it is worth to involve to the model also additional information about the relations among the weights. Unlike the previous two cases, as the fuzzy vector of weighted values UE is neither separable nor separable on some crisp relation, computation of the fuzzy weighted averages UEN and UER is not a problem of constraint fuzzy arithmetics. Taking into account the separability of WS3 on S 3 and of WRW on RW, for all a 2 (0, 1], the following holds

aNWS

NW a S

aNWR

3

3

;UE ð

;UE ð

W

aÞ ¼

aÞ ¼

;UE ð

aÞ ¼

min w1  u1 þ w2  u2 þ w3  u3 ;  Ni ðaÞ; i 2 N3 wi 2 ½wNi ðaÞ; w P3 i¼1 wi ¼ 1  i ðaÞ; i 2 N3 ui 2 ½ui ðaÞ; u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 ðu1  3Þ þ ðu2  6Þ2 þ 19 ðu3  6Þ2 6 1  a 4

ð96Þ

max w1  u1 þ w2  u2 þ w3  u3 ;  Ni ðaÞ; i 2 N3 wi 2 ½wNi ðaÞ; w P3 i¼1 wi ¼ 1  i ðaÞ; i 2 N3 ui 2 ½ui ðaÞ; u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 ðu1  3Þ þ ðu2  6Þ2 þ 19 ðu3  6Þ2 6 1  a 4

ð97Þ

min    Ni ðaÞ ; i 2 N3 wi 2 wNi ðaÞ; w P3 i¼1 wi ¼ 1

w1  u1 þ w2  u2 þ w3  u3 ;

w2  w1 P 0

ð98Þ

w1  w3 P 0  i ðaÞ; i 2 N3 ui 2 ½ui ðaÞ; u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 ðu1  3Þ þ ðu2  6Þ2 þ 19 ðu3  6Þ2 6 1  a 4 NW a R

W

;UE ð

aÞ ¼

max    Ni ðaÞ ; i 2 N3 wi 2 wNi ðaÞ; w P3 i¼1 wi ¼ 1 w2  w1 P 0 w1  w3 P 0  i ðaÞ; i 2 N3 ui 2 ½ui ðaÞ; u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 ðu1  3Þ þ ðu2  6Þ2 þ 19 ðu3  6Þ2 6 1  a 4

w1  u1 þ w2  u2 þ w3  u3 :

ð99Þ

In Fig. 6, we can see that the fuzzy numbers UEN and UER are narrower than UN and UR. Thus, involving the possible relations among the weighted values to the model can make the result also less uncertain. Now, let us focus on defuzzification of the fuzzy weighted averages UN, UR, UEN, and UER. Let us note that the marginal possibility distributions of particular weights and of particular weighted values are in all cases the same. Thus, we will see how the various interactions among the weights and among the weighted values affect the process of defuzzification. If we use for defuzzification the centers of gravity or the middle points of the mean intervals, i.e. the methods of defuzzification that depend only on the membership functions of the fuzzy numbers, we obtain the following results: GR(UN) = 5.036, GR(UR) = 5.016, GR(UEN) = 5.046 and GR(UER) = 5.03, and EðU N Þ ¼ 5:072; EðU R Þ ¼ 5:025; EðU EN Þ ¼ 5:098 and EðU ER Þ ¼ 5:065. We can see that all these characteristics are slightly different which means that the effect of the interactions among the variables on the membership functions is not symmetric. However, the ordering of the fuzzy weighted averages according to both characteristics is the same. On the contrary, defuzzification by means of the expected values of the fuzzy weighted averages shows us how the weighted average operation aN transfers the uncertainty of input variables to the output. The expected values of these fuzzy numbers are the following: E(UN) = E(UEN) = 5.135 and E(UR) = E(UER) = 5.104. The equalities E(UN) = E(UEN) and E(UR) = E(UER) are caused by the fact that the a-cuts Usepa and UEa are symmetric with respect to the central point uc = (3, 6, 6), i.e. for any u 2 R3 , it holds that if uc + u belongs to Usepa, or to UEa, then also uc  u belongs to Usepa, or to UEa, respectively. For the computation of UT, we can again use the fact that the fuzzy vector Usep is separable, and thus, we can employ the formulas (81) and (82). Moreover, since the fuzzy vector of weights WT is separable on T, the optimization problems look for all a 2 (0, 1] as follows:

O. Pavlacˇka / Information Sciences 181 (2011) 4969–4992

4987

Fig. 5. The fuzzy weighted averages UN and UR.

aW WT ;Usep ðaÞ ¼

W a WT ;Usep ðaÞ ¼

min

w1  u1 ðaÞ þ w2  u2 ðaÞ þ w3  u3 ðaÞ ; w1 þ w2 þ w3

ð100Þ

max

 1 ðaÞ þ w2  u  2 ðaÞ þ w3  u  3 ðaÞ w1  u : w1 þ w2 þ w3

ð101Þ

 i ðaÞ;i2N 3 wi 2½wi ðaÞ;w w1 þw2 w3 ¼1

 i ðaÞ;i2N 3 wi 2½wi ðaÞ;w w1 þw2 w3 ¼1

Because of nonseparability of UE, computation of UET is significantly more difficult. For all a 2 (0, 1], the following holds

aW WT ;UE ðaÞ ¼

min  i ðaÞ; i 2 N3 wi 2 ½wi ðaÞ; w

w1  u1 þ w2  u2 þ w3  u3 ; w1 þ w2 þ w3

w1 þ w2  w3 ¼ 1

ð102Þ

 i ðaÞ; i 2 N3 ui 2 ½ui ðaÞ; u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 ðu1  3Þ þ ðu2  6Þ2 þ 19 ðu3  6Þ2 6 1  a 4 W a WT ;UE ðaÞ ¼

max  i ðaÞ; i 2 N3 wi 2 ½wi ðaÞ; w w1 þ w2  w3 ¼ 1  i ðaÞ; i 2 N3 ui 2 ½ui ðaÞ; u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 ðu1  3Þ þ ðu2  6Þ2 þ 19 ðu3  6Þ2 6 1  a 4

w1  u1 þ w2  u2 þ w3  u3 : w1 þ w2 þ w3 ð103Þ

The membership functions of the fuzzy weighted averages UT and UET are shown in Fig. 7. We can see again that involving the additional condition concerning the weighted values affects the uncertainty of the result. As for the defuzzification of UT and UET, the characteristics depending only on the membership functions again slightly differs from each other, GR(UT) = 5.26, GR(UET) = 5.285, and EðU T Þ ¼ 5:172; EðU ET Þ ¼ 5:209, while since the applied fuzzy vectors of weights are the same, the expected values E(UT) and E(UET) are equal, E(UT) = E(UET) = 5.314. Example 7. Let us focus now on the problem of defuzzification of the fuzzy weighted average in more detail. Particularly, by this example, we will illustrate that the expected value of the fuzzy weighted average reflects also the interactions among the weights and among the weighted values that do not affect its membership function. Let the marginal possibility distributions of the couple of weighted values be given by the triangular fuzzy numbers  i ðaÞga2½0;1 ; i ¼ 1; 2. First, let us assume that there are no U1 = h0, 50, 100i, and U2 = h0, 150, 300i. Let us denote U i ¼ f½ui ðaÞ; u interactions between the weighted values and thus, the joint possibility distribution is expressed by the separable fuzzy vector Usep = U1  U2. Second, let the joint possibility distribution of the weighted values be expressed by the 2-dimensional  1 ðaÞ  ½u2 ðaÞ; u  2 ðaÞ that fuzzy vector UL, whose a-cuts ULa are given for all a 2 (0, 1] as the set of all vectors ðu1 ; u2 Þ 2 ½u1 ðaÞ; u satisfy the following inequalities

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Fig. 6. The fuzzy weighted averages UEN and UER.

Fig. 7. The fuzzy weighted averages UT and UET.



u1  u1 ðaÞ  1 ðaÞ  u1 ðaÞ u

2 6

u2  u2 ðaÞ 6  2 ðaÞ  u2 ðaÞ u

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u1  u1 ðaÞ :  1 ðaÞ  u1 ðaÞ u

ð104Þ

An a-cut of the fuzzy vector UL is shown in Fig. 8. Further, let uncertain normalized weights be expressed by the couple of normalized fuzzy weights W1 = h0, 0.5, 1i and W2 = h0, 0.5, 1i, i.e. the corresponding fuzzy vector of normalized weights is given by WS2 :¼ ðW 1  W 2 Þ \ S 2 . N Let U N :¼ aN F ðWS 2  Usep Þ, and U LN :¼ aF ðWS 2  UL Þ. As both the fuzzy vectors of weighted values Usep and UL satisfy for all a 2 (0, 1] the condition (83), it is obvious that UN = ULN. Since the fuzzy weighted averages UN and ULN have the same membership function (it is shown in Fig. 9), their centers of gravity, GR(UN) = GR(ULN) = 108.33, and the middle points of their mean intervals, EðU N Þ ¼ EðU LN Þ ¼ 116:64, are identical. However, as there are no interactions between the weighted values in the first case, while in the second case, the admissible combinations of values from the a-cuts UNa and ULNa are more concentrated towards (u1(a), u2(a)) than towards  1 ðaÞ; u  2 ðaÞÞ (see Fig. 8), the expected values of UN and of ULN are different, E(UN) = 100, and E(ULN) = 95.02. ðu Example 8. Let us show now that some special interactions among the weights and among the weighted values can cause that even though the weights and the weighted values are not crisp, the fuzzy weighted average is a real number. Let the joint possibility distribution of uncertain weighted values be expressed by the 4-dimensional fuzzy vector UQ U :¼ ðU 1  U 2  U 3  U 4 Þ \ Q U , where U1 = h1, 2, 5i, U2 = h5, 2, 1i, U3 = h1, 5, 6i, and U4 = h6, 5, 1i are the marginal possibility distributions, and the crisp relation QU,

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Fig. 8. An a-cut of the fuzzy vector of weighted values UL.

Q U :¼ fðu1 ; u2 ; u3 ; u4 Þ 2 R4 ju1 þ u2 ¼ 0; u3 þ u4 ¼ 0g; represents the set of all admissible combinations of weighted values. Further, let the joint possibility distribution of uncertain normalized weights be given by the 4-dimensional fuzzy vector of normalized weights WQ W :¼ ðW 1  W 2  W 3  W 4 Þ \ Q W , where W1 = W2 = h0, 0.1, 0.5i, and W3 = W4 = h0, 0.4, 0.5i are the marginal possibility distributions of particular weights, and the crisp relation QW,

Q W :¼ fðw1 ; w2 ; w3 ; w4 Þ 2 S 4 jw1 ¼ w2 ; w3 ¼ w4 g describes the set of all admissible combinations of the values of normalized weights. The fuzzy weighted average of UQ U with the fuzzy vector of normalized weights WQ W is the fuzzy number

U :¼ aNF ðUQ U  WQ W Þ ¼ f½aNUQ

U

;WQ

W

NU ðaÞ; a Q

U

;WQ

W

ðaÞga2½0;1 ;

where

aNUQ

U

;WQ

W

ðaÞ ¼

min wi 2 W ia ; i 2 N4 w1 ¼ w2 ; w3 ¼ w4

4 X

wi  ui ;

i¼1

ð105Þ

w1 þ w2 þ w3 þ w4 ¼ 1 ui 2 U ia ; i 2 N4 u1 þ u2 ¼ 0; u3 þ u4 ¼ 0

NU a Q

U

;WQ

W

ðaÞ ¼

max wi 2 W ia ; i 2 N4

4 X

wi  ui :

i¼1

w1 ¼ w2 ; w3 ¼ w4

ð106Þ

w1 þ w2 þ w3 þ w4 ¼ 1 ui 2 U ia ; i 2 N 4 u1 þ u2 ¼ 0; u3 þ u4 ¼ 0 It can be easily seen from (105) and (106) that aNUQ

U

;WQ

W

NU ðaÞ ¼ a Q

U

;WQ

W

ðaÞ ¼ 0 for all a 2 [0, 1], i.e. U = 0.

Remark 19. In practical case, it might happen that an expert describes the weights by nonnegative fuzzy numbers  i ðaÞga2½0;1 ; i ¼ 1; 2; . . . ; m, that satisfy with respect to W m the condition (16) but do not satisfy the condition W i ¼ f½wi ðaÞ; w  i ð1Þ > 0 for at least one i 2 Nm and there exists a 2 [0, 1] such that wi(a) = 0 for all i 2 Nm. In such a case, (58). It means that w the fuzzy set WW m ¼ ðW 1  W 2      W m Þ \ W m is not a fuzzy vector as WW m a are not closed for a 6 a0, where a0 = max{a 2 [0, 1]jwi(a) = 0 for all i 2 Nm}; the existence of such an a0 follows from the monotonicity and left-continuity of the functions wi, i 2 Nm.

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Fig. 9. The fuzzy weighted averages UN and ULN.

Although WW m is not a fuzzy vector, we can still compute according to the extension principle (5) the fuzzy set W m W AW F :¼ aF ðWW m  UÞ for any fuzzy vector of weighted values U 2 F V ðR Þ. Let us show now that AF is a fuzzy number. For W W W F ðaÞ, where aW W any a 2 (a0,1], the a-cut WW m a is a compact and convex subset of W m , and thus, AF a ¼ ½aF ðaÞ; a F ðaÞ and aF ðaÞ are obtained by

aW F ðaÞ ¼

min

Pm i¼1 wi  ui ðaÞ Pm ; i¼1 wi

ð107Þ

max

Pm  i¼1 wi  ui ðaÞ Pm : w i¼1 i

ð108Þ

 i ðaÞ;i2N m wi 2½wi ðaÞ;w ðu1 ;u2 ;...;um Þ2Ua

W a F ðaÞ ¼

 i ðaÞ;i2N m wi 2½wi ðaÞ;w ðu1 ;u2 ;...;um Þ2Ua

For a 2 [0, a0], let us denote

(

a

Nm :¼ and U a :¼

) 0

i 2 Nm ja 6 sup a

;

 i ða0 Þ>0 w

S

i2Nam ½Uia .

W W Then, AW F a ¼ ½aF ðaÞ; aF ðaÞ, where

W aW u and a u: F ðaÞ ¼ min F ðaÞ ¼ max a a u2U

ð109Þ

u2U

W As U is a fuzzy vector and W1, W2, . . . , Wm are fuzzy numbers, it follows from (107)–(109) that the functions aW F and aF are W W W W right-continuous at 0, left-continuous on (0, 1], and satisfy aW F ðaÞ 6 aF ðbÞ 6 aF ðbÞ 6 aF ðaÞ for all 0 6 a 6 b 6 1. Thus, AF is a fuzzy number.  i ða0 Þ > 0 for all i 2 Nm, there can exist a 2 [0, a0] and x 2 AW However, unless w F a such that there are no weights P m wi ui P ðw1 ; w2 ; . . . ; wm Þ 2 WW m a and no weighted values (u1, u2, . . . , um) 2 Ua satisfying x ¼ i¼1 . If it is in contradiction with m i¼1

wi

expert’s opinion, we can approximate the fuzzy number AW F by a fuzzy weighted average of U with some fuzzy vector on W m . Let us show now the possible way of such an approximation. For c 2 R satisfying

 i ð1Þ; 0 < c 6 max w

ð110Þ

i2N m

let us denote

( W mc :¼

ðw1 ; w2 ; . . . ; wm Þ 2 W m j

m X

) wi P c :

i¼1

Since W mc is a closed and convex set, and the condition (110) implies

ðCore W 1  Core W 2      Core W m Þ \ W mc – ;;

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the fuzzy set WW mc is a fuzzy vector on W m . Thus, we can compute the fuzzy number ACF :¼ aW F ðWW mc  UÞ employing the forC mulas (72)–(74). As WW mc  WW m , it holds that ACF # AW F . Thus, we can interpret the fuzzy number AF as an approximation of C AW F that is consistent with expert opinion since every value from some a-cut AF a is obtained as the weighted average of some weights from a-cuts of fuzzy weights and some weighted values that belong to Ua. It is obvious that the quality of such approximation depends on the choice of the value c. Generally, it holds that the smaller c is, the closer ACF is to AW F . Let us show now some examples how to set c in some particular cases. For instance, if for 0  i ða0 Þ > 0 for all i 2 Nm, then setting c 6 mini2Nm w  i ða0 Þ will ensure that ACFa ¼ AW some a0 2 ½0; 1; w F a for all a 6 a . Especially, if 00 00 00 a0 = 1, we obtain ACF ¼ AW . Further, if for some a 2 [ a , 1], w ( a ) > 0 for all i 2 N , then setting c 6 min 0 i m i2Nm wi ða Þ will F W 00  ensure that ACFa ¼ AW for all a P a . Finally, let us study the case when w ð a Þ > 0 for all i 2 N , i.e. when A is also in m 0 i Fa F consistence with expert’s opinion in the sense mentioned above. We may ask if there is some c > 0 such that ACF ¼ AW F . We have already found out in the first example that the equality holds if a0 = 1. For a0 < 1, we have to employ an obvious fact that the boundary values of ACFa are for all a > a0 reachable by such vector of weights, where wi = wi(a) for at least one i 2 Nm, and  j ðaÞ for at least one j 2 Nm, j – i. Let us denote wj ¼ w

N :¼ fi 2 Nm jwi ðaÞ > 0 for all a > a0 g:  j ðaÞ ¼ 0 for all a P a0 and for all j 2 NmnN⁄, then we obtain the desirable equality, if we set It can occur two cases. If w c 6 mini2N wi ð1Þ. Otherwise, ACF ¼ AW F if

c 6 inf

a>a0

X

!

wi ðaÞ þ

i2N 

min

 j ðaÞ–0 j2Nm nN  :w

 j ðaÞ : w

5. Conclusion The weighted average is one of the most frequently used aggregation operator in decision making models. Since the input variables–weights and weighted values–are in the models usually set expertly, their values are more or less uncertain. Up to now, uncertain general weights and uncertain weighted values were supposed to be modeled by noninteractive fuzzy numbers, and uncertain normalized weights or probabilities by a tuple of normalized fuzzy weights (with the exception of [26], where the fuzzy vectors of normalized weights were already considered). In the paper, we have shown that in comparison with this approach, applying the fuzzy vectors extends the possibilities of modeling the complete expert’s knowledge concerning the weights and weighted values. Besides describing the uncertain values of particular variables by fuzzy numbers representing the marginal possibility distributions, an expert can add another kind of information concerning the weights or weighted values in the form of a crisp relation that expresses all available combinations of their values (e.g. their ordering). It was shown that in such a case, the corresponding joint possibility distribution is expressed by a fuzzy vector separable on this crisp relation, and computation of the fuzzy weighted average is a matter of constrained fuzzy arithmetic. An expert has also the possibility to give even more complex information about the weights or weighted values by a fuzzy vector that is not separable on any crisp relation; computation of the fuzzy weighted average exceeds then the bounds of constrained fuzzy arithmetic. The advantage of the approach presented in the paper is that the resulting fuzzy weighted average is not overly imprecise since every available information about the variables is involved in computation. This fact has been illustrated by several examples. However, an obvious disadvantage is the possible increase of computational complexity. As for the further investigation, we can consider another t-norm instead of the minimum operator for modeling the Cartesian product of fuzzy numbers in (13) or (14). For unrelated variables, such idea was introduced in [5], and then, it was widely studied (see e.g. [6] and references therein). The a-cuts of the Cartesian product of fuzzy numbers that is modeled by a t-norm are in general not convex. Thus, we would have to admit within this concept that the fuzzy sets expressing uncertain input variables have not necessary be fuzzy vectors. If we want the result of the fuzzy extension of a continuous function to be still a fuzzy number, we can use the fact that Theorem 5 generally holds for a fuzzy set X 2 F ðRm Þ whose a-cuts are closed, bounded, and simply connected sets (see [31]). Therefore, the conditions for some mutual suitability of an m-tuple of fuzzy numbers, a crisp relation, and a t-norm should be derived so to ensure these properties. Applying this concept to modeling uncertain weights or weighted values would lead to the further decrease of uncertainty of the resulting fuzzy weighted average. However, a disadvantage would be another increase of computational complexity. In the end, it is worth noting that the fuzzy vectors can be also applied for modeling uncertain variables in other aggregation operators that are present in decision making models, e.g. OWA operator [35], etc. In fuzzy extensions of such aggregation operators studied so far, uncertain weighted values were modeled only by noninteractive fuzzy numbers, and uncertain normalized weights only by a tuple of normalized fuzzy weights (see e.g. [29] and references therein). Acknowledgements The author thanks the anonymous reviewers for their constructive comments and suggestions that were very helpful in improving the paper.

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References [1] S. Baas, H. Kwakernaak, Rating and ranking of multiple-aspect alternatives using fuzzy sets, Automatica 13 (1977) 47–58. [2] L.M. De Campos, J.F. Heute, S. Moral, Probability intervals: a tool for uncertain reasoning, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 2 (1994) 167–196. [3] W.M. Dong, F.S. Wong, Fuzzy weighted averages and implementation of the extension principle, Fuzzy Sets and Systems 21 (1987) 183–199. [4] D. Dubois, H. Prade, The use of fuzzy numbers in decision analysis, in: M.M. Gupta, E. Sanchez (Eds.), Fuzzy Information and Decision Processes, NorthHolland, Amsterdam, 1979, pp. 309–321. [5] D. Dubois, H. Prade, Additions of interactive fuzzy numbers, IEEE transactions on Automatic Control 26 (4) (1981) 926–936. [6] D. Dubois, E. Kerre, R. Mesiar, H. Prade, Fuzzy interval analysis, in: D. Dubois, H. Prade (Eds.), Fundamentals of fuzzy sets, Kluwer Academic Publishers, Boston, London, Dordrecht, 2000, pp. 483–582. [7] R. Fullér, P. Majlender, On weighted possibilistic mean and variance of fuzzy numbers, Fuzzy Sets and Systems 136 (2003) 363–374. [8] R. Fullér, P. Majlender, On interactive fuzzy numbers, Fuzzy Sets and Systems 143 (2004) 355–369. [9] R. Fullér, P. Majlender, Correction to: on interactive fuzzy numbers [Fuzzy Sets and Systems 143 (2004) 355–369], Fuzzy Sets and Systems 152 (2005) 159. [10] Y.-Y. Guh, C.C. Hong, K.M. Wang, E.S. Lee, Fuzzy weighted average: a max–min paired elimination method, Computers & Mathematics with Applications 32 (1996) 115–123. [11] Y.-Y. Guh, Ch.-Ch. Hon, E.S. Lee, Fuzzy weighted average: the linear programming approach via Charness and Cooper’s rule, Fuzzy Sets and Systems 117 (2001) 157–160. [12] E. Hisdal, Conditional possibilities independence and noninteraction, Fuzzy Sets and Systems 1 (1978) 283–297. [13] C.-H. Juang, X.H. Huang, D.J. Elton, Fuzzy information processing by Monte Carlo simulation technique, Civil Engineering and Environmental Systems 8 (1991) 19–25. [14] G.J. Klir, Y. Pan, Constrained fuzzy arithmetic: basic questions and some answers, Soft Computing 2 (1998) 100–108. [15] H. Kwakernaak, Fuzzy random variables – Vol. I: Definitions and theorems, Information Sciences 15 (1978) 1–29. [16] H. Kwakernaak, Fuzzy random variables – Part II: Algorithms and examples in discrete case, Information Sciences 17 (1979) 253–278. [17] t.S. Liou, M.J.J. Wang, D.J. Elton, Fuzzy weighted average: An improved algorithm, Fuzzy Sets and Systems 49 (1992) 307–315. [18] M. Mashinchi, An application of fuzzy set representation, Scientia Iranica 2 (4) (1996) 341–346. [19] S. Nahmias, Fuzzy variables in the random environment, in: M.M. Gupta, R. Ragade, R.R. Yager (Eds.), Advances in Fuzzy Set theory, North-Holland, Amsterdam, 1979, pp. 165–180. [20] C.V. Negoita, D.A. Ralescu, Representation theorems for fuzzy concepts, Kybernetes 4 (1975) 169–174. [21] Y. Pan, G.J. Klir, Bayesian inference based on interval probabilities, Journal of Intelligence and Fuzzy Systems 5 (1997) 193–203. [22] Y. Pan, B. Yuan, Bayesian inference of fuzzy probabilities, International Journal of General Systems 26 (1–2) (1997) 73–90. [23] O. Pavlacˇka, Fuzzy methods of decision making, Dissertation thesis, Palacky´ University Olomouc, 2007 (in Czech). [24] O. Pavlacˇka, J. Talašová, The fuzzy weighted average operation in decision making models, in: Proceedings of the 24th International Conference ˇ , 13th–15th September 2006, Plzen ˇ 2006, pp. 419–426. Mathematical Methods in Economics, Plzen [25] O. Pavlacˇka, J. Talašová, Application of the Fuzzy Weighted Average of Fuzzy Numbers in Decision Making Models, in: U. Bodenhofer, V. Novák, M. Šteˇpnicˇka (Eds.), New Dimensions in Fuzzy Logic and Related technologies, Proceedings of the 5th EUSFLAT Conference, Ostrava, Czech Republic, Ostravská univerzita, Ostrava, vol. II, September 11–14, 2007, pp. 455–462. [26] O. Pavlacˇka, J. Talašová, Application of Fuzzy Vectors of Normalized Weights in Decision Making Models, in: J.P. Carvalho, D. Dubois, U. Kaymak and J.M.C. Sousa (Eds.), Proceedings of the Joint 2009 IFSA World Congress and 2009 EUSFLAt Conference, Lisbon, Portugal, July 20–24, 2009, pp. 495–500. [27] O. Pavlacˇka, J. Talašová, Fuzzy vectors as a tool for modeling uncertain multidimensional quantities, Fuzzy Sets and Systems 161 (2010) 1585–1603. [28] J. Talašová, NEFRIt – multicriteria decision making based on fuzzy approach, Central European Journal of Operations Research 8 (4) (2000) 297–319. [29] J. Talašová, I. Bebcˇáková, Fuzzification of aggregation operators based on Choquet integral, Aplimat – Journal of Applied Mathematics 1 (1) (2008) 463– 474. [30] J. Talašová, O. Pavlacˇka, Fuzzy probability spaces and their applications in decision making, Austrian Journal of Statistics 35 (2–3) (2006) 347–356. [31] R. Viertl, Statistical Methods for Non-Precise Data, CRC Press, Boca Raton, Florida, 1996. [32] Y.-M. Wang, t. M.S. Elhag, On the normalization of interval and fuzzy weights, Fuzzy Sets and Systems 157 (2006) 2456–2471. [33] P. Walley, Statistical Reasoning with Imprecise Probabilities, Chapman and Hall, London, 1991. }hlmann, A Methodology for Uncertainty in Knowledge-based Systems, Springer-Verlag, New York, 1990. [34] K. Weichselberger, S. Po [35] R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE transactions on Systems, Man, and Cybernetics 3 (1) (1988) 183–190. [36] L.A. Zadeh, Concept of a linguistic variable and its application to approximate reasoning I, II, Information Sciences 8 (1975) 199–249, 301–357; III, Information Sciences 9 (1975) 43–80.