On various approaches to normalization of interval and fuzzy weights

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ScienceDirect Fuzzy Sets and Systems 243 (2014) 110–130 www.elsevier.com/locate/fss

On various approaches to normalization of interval and fuzzy weights Ondˇrej Pavlaˇcka Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University Olomouc, 17. listopadu 12, 771 46 Olomouc, Czech Republic Received 5 October 2012; received in revised form 11 July 2013; accepted 26 July 2013 Available online 14 August 2013

Abstract The paper deals with the problem of fuzzification of the procedure of normalization of weights. First, the existing methods for normalization of interval and fuzzy weights are reviewed. Second, we study the problem of normalization of a fuzzy vector of weights that expresses the joint possibility distribution of initial weights. We show that a correct way is to apply the extension principle proposed by Zadeh, since the result of such normalization is the fuzzy vector of normalized weights that expresses the true joint possibility distribution of normalized weights. Further, we establish some properties of this approach to normalization that are important from the point of view of real applications. Finally, since an n-tuple of non-interactive interval or fuzzy weights can be viewed as a fuzzy vector of weights of a special kind, we investigate normalization of such kind of fuzzy vectors of weights according to the extension principle. We show that from the point of view of the way of modelling uncertain normalized weights, the result of this approach can be directly compared only with the result of normalization proposed by Wang and Elhag (2006) [35]. We find out that it is not sufficient to express the result of normalization only by an n-tuple of normalized interval or fuzzy weights together with the constraint that the sum of the weights is equal to 1, since it can cause a false increase of uncertainty in the model. This fact is illustrated by an example. © 2013 Elsevier B.V. All rights reserved. Keywords: Normalization of weights; Interval weights; Fuzzy weights; Fuzzy vector; Fuzzy weighted average; Normalized fuzzy weights

1. Introduction In multiple criteria decision making (MCDM) models, weights usually represent some kind of ordinal or cardinal information about the importance of criteria. The weights are in general expressed by nonnegative real numbers whose sum is different from zero. Particularly, if the sum of the weights is equal to 1, they are called normalized weights. Throughout the paper, we will denote the set of all n-tuples of general weights by Wn , i.e.  n Wn := R+ \ {o}, 0 where o denotes the zero vector, and the set of all n-tuples of normalized weights, the n-dimensional probability simplex, by Sn , i.e. E-mail address: [email protected]. 0165-0114/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2013.07.026

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 n    Sn := (v1 , . . . , vn ) ∈ R vi  0, i = 1, . . . , n, vi = 1 . n

i=1

In MCDM models, the weights of criteria are often ill-known. Their values are commonly set subjectively on the basis of experts’ experiences or opinions. We can also face incomplete or missing information about the importance of criteria. Such kinds of uncertainty can be sufficiently modelled by means of tools of fuzzy sets theory. In the literature, uncertain non-normalized weights are usually expressed (see e.g. [1,5,14,15,17,20,25,32,35]) by a tuple of non-interactive non-negative fuzzy numbers. For describing uncertain normalized weights, it was shown in [24,25,33,35] that a special structure of interactive fuzzy numbers on [0, 1] called a tuple of normalized fuzzy weights has to be applied because of the normalization condition. Construction of this structure is based on the assumption that only such combinations of values of particular weights whose sum is equal to 1 are admissible. The identical structure of fuzzy numbers was in [22] called a feasible tuple of fuzzy probabilities. It represented a generalization of a tuple of reachable interval probabilities introduced e.g. in [4,21,37]. A different structure of intervals or fuzzy numbers is proposed for modelling uncertain values of normalized weights in [32]. Normalized interval and fuzzy weights are treated as intervals or fuzzy objects, not as only constrains on standard (real valued) normalized ones. A more general approach to modelling uncertain weights was proposed in [23,28]. It was shown that uncertain n-tuples of non-normalized or normalized weights can be appropriately expressed by n-dimensional fuzzy vectors representing joint possibility distributions of the weights. In comparison to the above mentioned n-tuples of non-interactive fuzzy weights or n-tuples of normalized fuzzy weights, the fuzzy vectors significantly extend the possibilities of utilizing the vague expert information concerning the weights. General weights often need to be normalized in MCDM models for the purpose of elimination of their dimensions. For instance, in Analytic Hierarchy Process (AHP) [30], the weight vector obtained from a pairwise comparison matrix has to be normalized so that we can aggregate the local weights in a hierarchical structure into a global weight vector. Such kind of normalization of an n-tuple of general weights to the corresponding n-tuple of normalized weights is described by the real-vector-valued function n : Wn → Sn defined for all (w1 , . . . , wn ) ∈ Wn in the following way: 

w1 wn n(w1 , . . . , wn ) := n , . . . , n . (1) i=1 wi i=1 wi For the same reason, there is also often a necessity to normalize interval or fuzzy weights in MCDM models under uncertainty, especially in AHP with interval or fuzzy judgements (see e.g. [2,3,36,38]). Thus, the normalization function given by (1) has to be extend to the case of uncertain weights. In the literature, several methods for normalization of an n-tuple of interval or fuzzy weights were proposed up to now (see [32,35] and references therein). The methods are based on different understanding of what does the normalization mean when interval or fuzzy weights are available. Normalization of fuzzy vectors of weights has not been considered in the literature up to now, it was studied only partly in [23]. The aim of the paper is to review the existing methods for normalization of interval and fuzzy weights and to study the problem of the correct extension of the normalization procedure given by (1) to the case when uncertain weights are expressed by fuzzy vectors. The paper is organized as follows. The existing methods to normalization of interval and fuzzy weights are discussed in Section 2. Special attention is given to the different understanding of correctness of the normalization procedure in particular methods. In Section 3, modelling of n-tuples of uncertain weights by fuzzy vectors is briefly described. The approach to normalization of weights that is based on the extension principle and that makes it possible to normalize also the fuzzy vectors of weights is studied in Section 4. Some important properties of this way of normalization of fuzzy vectors of weights are shown in Section 5. Since an n-tuple of non-interactive interval or fuzzy weights can be expressed by a fuzzy vector of weights of a special kind, its normalization according to the extension principle is studied separately in Section 6. Finally, some concluding remarks are given. 2. Existing approaches to normalization of interval and fuzzy weights In this section, we will summarize the existing approaches to normalization of interval and fuzzy weights. We will discuss especially well-foundedness of the criteria of correctness of the normalization procedure that have been considered in the literature.

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Fuzzy weights are usually normalized in such a way that the chosen method for normalization of interval weights is applied to α-cuts of fuzzy weights. Thus, for the sake of simplicity, we will analyze further in this section only the methods of normalization of positive initial interval weights [w i , wi ], i = 1, . . . , n. In the case of real-valued weights, the normalization given by (1) provides the weights whose sum is equal to 1. In the interval setting, the following problem arises: Let us suppose that by the normalization procedure we obtain the n-tuple of interval weights [v i , v i ], i = 1, . . . , n. It is clear that unless the obtained intervals are degenerated (v i = v i , i = 1, . . . , n) or at least one of the intervals is inverted (v i > v i ), their sum according to conventional interval arithmetic  n n n    [v i , v i ] = vi , vi (2) i=1

i=1

i=1

can never be equal to [1, 1]. Thus, it is necessary to formulate the properties of the interval normalization. 2.1. Normalization proposed by Chang and Lee [3] Chang and Lee [3] proposed the following formulation of the normalization condition for the obtained intervals:

n  n    vj · v j = 1. (3) j =1

j =1

Chang and Lee [3] further showed that this condition is always satisfied when the normalization method based on conventional interval or fuzzy arithmetic is considered. By this method, the initial interval weights are normalized A into the normalized interval weights [v A i , v i ], i = 1, . . . , n, as follows:    A A wi wi [w i , wi ] = n v i , v i = n , n , i = 1, . . . , n. (4) j =1 [w j , w j ] j =1 w j j =1 w j However, Wang and Elhag [35] showed that there is no evidence to support the assumption given by (3) for normalized interval and fuzzy weights. Let us also note that intervals obtained by (4), and thus satisfying (3), do not have to be necessarily subsets of the unit interval [0, 1] that represents the natural range for the values of particular normalized weights. 2.2. Normalization proposed by Wang and Elhag [35] Another approach to normalization was introduced by Wang and Elhag [35]. It is based on the assumption that normalized interval (or fuzzy) weights express the ranges of the particular weights that are required to be summed to one. This means that for any computation with such interval weights, the concept of constrained interval (or fuzzy) arithmetic (see e.g. [18]) has to be applied. Hence, summing such interval (or fuzzy) weights according to the conventional interval (or fuzzy) arithmetic (2) is not correct. Let V = ([v 1 , v 1 ], . . . , [v n , v n ]) be a vector of interval weights such that 0  v i  v i , i = 1, . . . , n. On the basis of these interval weights, Wang and Elhag construct the set of normalized weight vectors    NV := (v1 , . . . , vn ) ∈ Sn  v i  vi  v i , i = 1, . . . , n , (5) and formulate the definition of normalization of interval weights as follows: The interval weight vector V is said to be normalized if and only if the set NV satisfies the following two conditions (note that the second one implies the first one): (*) There exists at least one normalized weight vector in the set NV , i.e. NV is non-empty. (**) The bounds v i and v i , i = 1, . . . , n, are all attainable in NV , i.e. both v i and v i represent the i-th component of at least one normalized weight vector belonging to NV . The weaker condition (*) is obviously satisfied if and only if

O. Pavlaˇcka / Fuzzy Sets and Systems 243 (2014) 110–130 n 

vi  1

and

i=1

n 

v i  1.

113

(6)

i=1

The condition (**) is satisfied (see [35]) if and only if n 

v j + max (v i − v i )  1 and

j =1

i=1,...,n

n 

v j − max (v i − v i )  1.

j =1

i=1,...,n

(7)

Let us note that unlike the property (3), the condition (**) ensures that the normalized interval weights [v i , v i ], i = 1, . . . , n, are subsets of [0, 1]. A Wang and Elhag [35] showed that the interval weights [v A i , v i ], i = 1, . . . , n, obtained by (4) do not fulfill in general the condition (7), and introduced the model of normalization that produces the normalized interval weight vector V. They distinguished two cases – independent and dependent initial interval weights. 2.2.1. Case of independent initial interval weights If the initial interval weights [w i , wi ], i = 1, . . . , n, are independent of each other, i.e. if the set of all initial non-normalized weight vectors is given as follows:    W := (w1 , . . . , wn ) ∈ Wn  w i  wi  wi , i = 1, . . . , n , they propose for normalization the following two formulas that were first developed by Dubois and Prade [9]: wi wi n n vi = min = , i = 1, . . . , n, wj ∈[wj ,w j ], j =1,...,n w w + i j =1 j j =1, j =i w j vi =

max

wj ∈[wj ,w j ], j =1,...,n

wi n

j =1 wj

=

wi +

wi n

j =1, j =i

wj

,

i = 1, . . . , n.

(8) (9)

Wang and Elhag [35] established that the obtained intervals satisfy the inequalities in (7), and assumed that the corresponding set of normalized weight vectors is the set NV given by (5). Let us discuss now the correctness of this model in detail. Obviously, [v i , v i ], i = 1, . . . , n, represent the projections of NV . Since wi /( nj=1 wj ), i = 1, . . . , n, are continuous functions on Wn , it follows from (8) and (9) that, for any i ∈ {1, . . . , n},    wi  [v i , v i ] = vi ∈ [0, 1]  vi = n , w j  wj  wj , j = 1, . . . , n . j =1 wj Thus, [v i , v i ], i = 1, . . . , n, represent also the projections of the set of normalized weight vectors  

 w1 wn  n V := , . . . , n ∈ Sn  wi  wi  wi , i = 1, . . . , n , i=1 wi i=1 wi

(10)

i.e. they express the true ranges of components of normalized weight vectors that are obtained by normalizing the weight vectors from W . Let us focus on the relation between the sets V and NV . According to [29], the equality V = NV holds in general only for n = 2. In the case of n  3, for the initial non-degenerated non-normalized interval weights such that w i > 0 for at least one i ∈ {1, . . . , n}, the set V is only a strict subset of NV . This means that NV contains also the normalized weight vectors that cannot be obtained by normalizing of any initial weight vector from W . Therefore, if n  3, it is not well-founded to consider the set of normalized weight vectors NV to be the result of the normalization procedure. The problem is illustrated by the following example. Example 1. Let us consider three interval weights W1 = [0.4, 0.6], W2 = [0.3, 0.6], and W3 = [0.2, 0.8], and let us assume that they are independent of each other, i.e. they generate the following set of initial non-normalized weight vectors:    W = (w1 , w2 , w3 ) ∈ W3  0.4  w1  0.6, 0.3  w2  0.6, 0.2  w3  0.8 . Applying the formulas (8) and (9), we obtain a triple of normalized interval weights V1 = [0.222, 0.545], V2 = [0.176, 0.5], and V3 = [0.143, 0.533] that within the approach by Wang and Elhag [35] generate the following set of normalized weight vectors:

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   NV = (v1 , v2 , v3 ) ∈ S3  0.222  v1  0.545, 0.176  v2  0.5, 0.143  v3  0.533 . However, the set NV contains for instance the normalized weight vector v = (0.545, 0.176, 0.279) that cannot be obtained by normalizing the weight vectors from W , i.e. there is no vector of non-normalized weights w ∈ W such that n(w) = (0.545, 0.176, 0.279). 2.2.2. Case of dependent initial interval weights The second case studied by Wang and Elhag [35] was normalization of the initial dependent interval weights [v i , v i ] ⊆ [0, 1], i = 1, . . . , n, that describe the ranges of particular weights whose sum is equal to one, but that satisfy only the weaker condition (6); the condition (7) is violated. This means that the corresponding interval weight vector V = ([v 1 , v 1 ], . . . , [v n , v n ]) generates the non-empty set of normalized weight vectors   n    n   NV := (v1 , . . . , vn ) ∈ R v  vi  v , i = 1, . . . , n, vi = 1 , i

i

i=1

v i

V

v i

but is not normalized because not all bounds and are attainable in NV . In such a case, “normalization” consists in finding the projections of the set NV , since they represent the true ranges of particular normalized weights. This kind of normalization of interval weights is not a generalization of the normalization procedure given by (1). For updating the bounds of the interval weights [v i , v i ], i = 1, . . . , n, Wang and Elhag [35] propose the following formulas:   n    vi = min vi = max v i , 1 − v j , i = 1, . . . , n, (11)   vj ∈[v j ,v j ], j =1,...,n n j =1 vj =1

vi =

max

vj ∈[v j ,v j ], j =1,...,n n j =1 vj =1

j =1, j =i

 vi = min

v i , 1 −

n 

 v j

,

i = 1, . . . , n.

(12)

j =1, j =i

This model is a particular case of the possibility distribution model proposed by Dubois and Prade [8]. The same formulas were also developed by De Campos et al. [4] for the case of imprecise probability weights. The obtained interval weights [v i , v i ], i = 1, . . . , n, satisfy (7). Since   n   n NV = (v1 , . . . , vn ) ∈ R  v i  vi  v i , i = 1, . . . , n, vi = 1 , i=1

[v i , v i ], i = 1, . . . , n, express the true ranges of the particular normalized weights. Therefore, this kind of normalization procedure is fully correct. 2.3. Normalization proposed by Sevastjanov et al. [32] Completely different approach to normalization was introduced by Sevastjanov et al. [32]. It is based on the treatment of normalized interval and fuzzy weights as interval and fuzzy objects, not as constraints on standard (real valued) normalized ones. From this point of view, Sevastjanov et al. [32] consider the condition (7) to be too restrictive and not to form an exhaustive set of possible desirable properties of the interval weights normalization. This assertion was in [32] illustrated by the following example which was taken from [19]: Let us consider three interval weights W1 = [0.4, 0.6], W2 = [0.3, 0.6], and W3 = [0.2, 0.8] that satisfy (6), but violate (7). From Eqs. (11) and (12), the normalized intervals V1 = [0.4, 0.5], V2 = [0.3, 0.4], and V3 = [0.2, 0.3] are obtained. This result is according to Sevastjanov et al. [32] counterintuitive for the following two reasons. Firstly, such a drastic reduction of the width of the widest interval W3 may not be accepted by decision makers. Secondly, W3 is according to Sevastjanov et al. the maximal weight in the interval sense before normalization, but normalized V3 is the minimal one. However, these arguments are not valid. As it is described above, the formulas (11) and (12) are suggested to be applied for normalization only in the specific

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case – when the initial interval weights describe the ranges of values that are required to be summed to 1. This means that the initial interval weights W1 , W2 and W3 in fact generate the set of normalized weight vectors   3   3 N(W ,W ,W ) = (v1 , v2 , v3 ) ∈ R vi ∈ Wi , i = 1, 2, 3, vi = 1 . 1

2

3

i=1

The obtained normalized interval weights V1 , V2 and V3 then express the projections of N(W1 ,W2 ,W3 ) . It can be easily seen (it follows from (12)) that the value of the third component of normalize weight vectors from N(W1 ,W2 ,W3 ) cannot be greater than 0.3, so the drastic reduction of the width of the widest interval W3 is in this example necessary and correct. As for the second reason, W3 is not the maximal weight in the interval sense since its lower bound 0.2 is lesser than the lower bounds of W1 and W2 . In fact, W3 is incomparable with W1 and W2 . So the ordering of weights is not reversed after normalization. Let us note that if we consider the initial interval weights to be non-interactive and employ the formulas (8) and (9) for their normalization (we will obtain the normalized interval weights V1 = [0.222, 0.545], V2 = [0.176, 0.5], and V3 = [0.143, 0.533], see Example 1), the third normalized interval weight V3 will be the widest one. Instead of the condition (7), if the interval and fuzzy normalized weights are treated as interval and fuzzy objects, Sevastjanov et al. [32] consider the following three properties of normalization to be desirable and natural ones. As the sum of normalized real-valued weights is always equal to 1, the sum of normalized interval or fuzzy weights should be equal to “near 1”. The expression “near 1” is suggested to be expressed by an interval or triangular fuzzy number centered around 1. The second desirable property of the interval or fuzzy weights normalization is derived from the fact that in the precise case, the process of normalization preserves the ratios of initial nonzero weights, i.e. for any (w1 , . . . , wn ) ∈ Wn such that wi > 0 and wj > 0 for some i, j ∈ {1, . . . , n}, it holds that wi = wj

nwi k=1 w n j k=1

wk wk

=

vi . vj

(13)

Sevastjanov et al. [32] require the ratios of means of normalized intervals or fuzzy weights to be “as close as possible” to those of initial intervals or fuzzy weights. The third desirable property consists in the fact that the ratios of the lengths of obtained normalized intervals should be “close enough” to those before normalization. According to Sevastjanov et al. [32], such a property is intuitively obvious. However, this property seems to be not well-founded. If the initial interval weights are independent of each other, it is clear that after normalization, the ordering of the lengths of the obtained normalized interval weights should not be reversed. But there is no reason why should normalization retain the ratios of the lengths of initial interval weights in general. The ratio of the lengths of two initial interval weights is not affected by the lengths of the other initial weights, while the ratio of the lengths of two normalized interval weights is. It follows from the fact that the i-th normalized weight is in the precise case given by vi = nwi w , which implies that in the case of interval j =1

j

weights, the length of the i-th normalized interval weight is affected also by the lengths of all other initial interval weights. Let mi = (w i + wi )/2, li = wi − w i , i = 1, . . . , n, be the means and lengths of non-normalized interval weights and m ˆ i and lˆi be the means and lengths of the corresponding normalized interval weights. Sevastjanov et al. [32] suggest the following aggregated measures of closeness of ratios of the means and lengths of non-normalized interval weights to ratios of the means and lengths of normalized interval weights:   

n  n  1 m ˆi 2 mi σm = − , (14) · n(n − 1) mj m ˆj i=1 j =1, j =i   

n  n  1 lˆi 2 li − . (15) σl = · n(n − 1) lj lˆj i=1 j =1, j =i

However, in illustrative numerical examples shown in [32], the following slightly different formulas were applied:

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n  n  1 m ˆi 2 mi  σm = − , · n(n − 1) mj m ˆj i=1 j =1, j >i   

n  n  1 lˆi 2 li  − . · σl = n(n − 1) lj lˆj i=1 j =1, j >i The first mentioned property serves as the basis of a definition of a normalized interval weight vector in [32]. The vector of interval weights [v i , v i ], i = 1, . . . , n, is said to be normalized if  n n n    [v i , v i ] = vi , v i = [1 − ε, 1 + ε], ε  1, (16) i=1

i=1

i=1

and if w i → wi for all i ∈ {1, . . . , n}, then ε → 0, i.e.

n  [v i , v i ] → 1.

(17)

i=1

Let us note that in contrast to the two previous definitions of normalized interval weights that were focused on the way of expressing uncertain normalized weights by intervals, this definition is rather a definition of the correctness of the normalization procedure, since the condition (17) is focused on the method of computing the normalized interval weights from initial non-normalized one. For normalization of interval weights [w i , wi ], i = 1, . . . , n, Sevastjanov et al. [32] propose to find an interval normalization factor [x, x] such that the interval weights  S S v i , v i = [wi , wi ] · [x, x] = [w i · x, wi · x], i = 1, . . . , n, (18) satisfy the conditions (16) and (17). On the basis of the so-called “interval extended zero” method [31] that was developed to solve interval and fuzzy equations, they established the following interval normalization factor: x = n

1

j =1 w j

where

n

ymin =

−

j =1 w j n j =1 w j

ymax + ymin , 2 nj=1 w j −

1

j =1 w j

−

ymax + ymin , 2 nj=1 wj

(19)

n

n

j =1 w j n + j =1 wj

x = n

,

j =1 w j

ymax = 1 − n

j =1 w j

.

(20)

Further, it was shown in [32] that n  

   v Si , v Si = 1 − εS , 1 + εS ,

i=1

where ymin + ymax . (21) 2 Finally, Sevastjanov et al. [32] made the comparison of their approach with the previously described methods of normalization. The comparison consisted in analyzing the results of normalization of six initial non-normalized interval weight vectors. It was shown that normalized interval weights [v Si , v Si ], i = 1, . . . , n, given by (18) do not satisfy in general the condition (7), but on the other hand, they are the only normalized interval weights whose sum according to the conventional interval arithmetics is centered around 1. As for the measures σm and σl , it was shown that in all considered examples, normalization given by (18) provided smaller values of σm and σl than the other methods. However, such a comparison seems to be disputable since each method of normalization is based on completely different ideas. As it was mentioned earlier, the normalized interval weights [v Si , v Si ], i = 1, . . . , n, are treated as independent intervals, they are not supposed to generate a set of normalized weight vectors. Thus, the condition (7) is of no avail for them. On the other hand, the normalized interval weights obtained by the formulas (8) εS =

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and (9), or (11) and (12), respectively, are interactive. They represent the true ranges of values whose sum is equal to 1. Therefore, summing these interval weights directly according to the conventional interval arithmetics is not valid. And from the same reason, comparing the ratios of their means and lengths is meaningless as well. Especially in the case of the normalization method given by (11) and (12) that consists in updating the bounds of initial dependent interval weights so that they can be attainable. The change of the ratios of their means and lengths is fully understandable here. 2.4. Normalization proposed by Jiménez et al. [16] At the end of this section, let us describe a method for normalization of interval weights that was considered by Jiménez et al. [16]. The normalized interval weights [v Ji , v Ji ], i = 1, . . . , n, are computed by means of the following formulas: ki · w i v Ji = , i = 1, . . . , n, (22) (w i + wi )/2 ki · wi v Ji = , i = 1, . . . , n, (23) (w i + wi )/2 where wi + wi , j =1 (w j + w j )

ki = n

i = 1, . . . , n.

Wang and Elhag [35] showed that this normalization method is based on the means of interval weights, since the formulas (22) and (23) can be rewritten as follows: wi , i = 1, . . . , n, (24) v Ji = n j =1 (w j + w j )/2 wi , j =1 (w j + w j )/2

v Ji = n

i = 1, . . . , n.

(25)

Wang and Elhag [35] also showed that n  

 v Jj + v Jj ≡ 2.

(26)

j =1

Jiménez et al. [16] did not discuss any special interpretation of the normalized interval weights obtained by (22) and (23). It can be easily seen from Eqs. (24) and (25) that [v Ji , v Ji ], i = 1, . . . , n, always satisfy the weaker condition (6). But they satisfy the stronger condition (7) if and only if n  j =1,j =i

wj + wi 

n 

wj + wi ,

for all i ∈ {1, . . . , n}.

j =1, j =i

Hence, they cannot represent in general the true ranges of the values whose sum is supposed to be equal to 1. However, an interesting result will be achieved if we compare this normalization procedure with the approach proposed by Sevastjanov et al. [32]. From (24) and (25), it follows that the sum of the normalized intervals is centered around 1: n n   n    J J  i=1 w i i=1 w i , n = 1 − εJ , 1 + εJ , v i , v i = n j =1 (w j + w j )/2 j =1 (w j + w j )/2 i=1

where

n wi − ni=1 w i (20) n ε J = i=1 = ymin . n i=1 w i + i=1 w i

(27)

From (27), it can be easily seen that if wi → wi for all i ∈ {1, . . . , n}, then ε J → 0. Hence, the normalization procedure given by (22) and (23) satisfy the conditions (16) and (17). Moreover, since ymin  ymax , it follows from (21) that

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ε J  ε S , i.e. ni=1 [v Ji , v Ji ] is never wider than ni=1 [v Si , v Si ]. Further, let m ˆ Ji and lˆiJ be the means and lengths of [v Ji , v Ji ], i = 1, . . . , n. Then, for all i, j ∈ {1, . . . , n}, the following equalities follow from (24) and (25): m ˆ Ji m ˆ Jj

=

n wi +w i /2 k=1 (w k +w k )/2 w j +w j n /2 k=1 (w k +w k )/2

=

(w i + wi )/2 mi , = (w j + wj )/2 mj

and lˆiJ = lˆJ j

n w i −w i k=1 (w k +w k )/2 w −w j n j k=1 (w k +w k )/2

=

wi − wi li = . wj − wj lj

Thus, the aggregated measures of closeness σm and σl given by (14) and (15) are in the case of [v Ji , v Ji ], i = 1, . . . , n, always equal to 0. Therefore, from the point of view of the desirable properties of interval weights normalization proposed by Sevastjanov et al. [32], we can say that the interval normalization factor [x, x], where x = n (w1 +w )/2 , i i i=1 provides substantially better results than the interval normalization factor given by (19). The problem is illustrated by the following example. Example 2. Let us consider the same three initial interval weights W1 = [0.4, 0.6], W2 = [0.3, 0.6], and W3 = [0.2, 0.8] as in Example 1. Let us compare now the results obtained by the normalization methods proposed by Sevastjanov et al. [32] and Jiménez et al. [16]. According to (18), [v S1 , v S1 ] = [0.238, 0.439], [v S2 , v S2 ] = [0.178, 0.439], and [v S3 , v S3 ] = [0.119, 0.586]. From (22) and (23), we obtain [v J1 , v J1 ] = [0.276, 0.414], [v J2 , v J2 ] = [0.207, 0.414], and [v J3 , v J3 ] = [0.138, 0.552]. First, let us compare the sums of the obtained normalized interval weights: ni=1 [v Si , v Si ] = [0.535, 1.465] and n J J i=1 [v i , v i ] = [0.621, 1.379]. We can see that both sums are intervals centered around 1, but the second one is S narrower; ε = 0.465 and ε J = 0.379. As for the other two desirable properties, the values of aggregated measures of closeness are in the first case the following: σmS = 0.012 and σlS = 0.127, while in the second case, both σmJ and σlJ are equal to 0, because the ratios of the means and lengths of the initial non-normalized interval weights are retained. In the next part of the paper, we will study the extension of the normalization procedure given by (1) to the case where uncertain weights are modelled by fuzzy vectors. 3. Fuzzy vectors of weights In this section, let us briefly introduce how can be n-tuples of uncertain weights modelled by fuzzy vectors. First, let us recall definitions and interpretations of fuzzy numbers and fuzzy vectors that will be considered further in the paper. 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(U ). By Core A and Supp A, we denote a core of A, i.e. Core A := {u ∈ U | A(u) = 1}, and a support of A, i.e. Supp A := {u ∈ U | A(u) > 0}, respectively. For any α ∈ (0, 1], Aα means an α-cut of A, i.e. Aα := {u ∈ U | A(u)  α}. Any crisp set A ⊆ U can be viewed as a fuzzy set of a special kind, where Aα = A holds for all α ∈ (0, 1]. Fuzzy numbers were introduced for expressing uncertain values of continuous variables; they can be interpreted as possibility distributions (see e.g. [13]). A fuzzy number is a fuzzy set X on the real axis R that fulfills the following conditions: (a) Core X = ∅, (b) for all α ∈ (0, 1], the α-cuts Xα are closed intervals, (c) Supp X is bounded. If the closure of Supp X, Cl(Supp X), is a subset of Q, where Q ⊂ R (usually a closed interval), then X is said to be a fuzzy number on Q. The family of all fuzzy numbers on Q will be denoted by FN (Q). Any fuzzy number X can be uniquely determined (see e.g. [7,10–12]) by a couple of functions x : [0, 1] → R and x : [0, 1] → R that describe the minimal and maximal values of the α-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 (this was proved in [27]), and satisfy

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119

x(α)  x(β)  x(β)  x(α) for all 0  α < β  1. In the sequel, the notation X = {[x(α), x(α)]}α∈[0,1] will be used for the fuzzy number X such that Xα = [x(α), x(α)] for all α ∈ (0, 1], and Cl(Supp X) = [x(0), x(0)]. Let us note that a real number as well as a closed interval can be viewed as fuzzy numbers of a special kind. A fuzzy number X = {[x(α), x(α)]}α∈[0,1] represents a real number c if x(α) = x(α) = c for all α ∈ [0, 1]. This fact enables us to properly handle precisely known values in fuzzy models. Furthermore, a closed interval [a, b], a < b, can be interpreted as a fuzzy number X = {[x(α), x(α)]}α∈[0,1] , where x(α) = a and x(α) = b for all α ∈ [0, 1]. A fuzzy vector is used for expressing the joint possibility distribution of an n-tuple of continuous variables (see e.g. [27] and references therein). An n-dimensional fuzzy vector X is a fuzzy set on Rn that fulfills the following conditions: (a) Core X = ∅, (b) for all α ∈ (0, 1], Xα are compact and convex subsets of Rn , (c) Supp X is bounded. If Cl(Supp X) ⊆ Q, where Q ⊂ Rn , then X is said to be a fuzzy vector on Q. The family of all n-dimensional fuzzy vectors on Q will be denoted by FV (Q). For any i ∈ {1, . . . , n}, the i-th projection of the n-dimensional fuzzy vector X is the fuzzy set [X]i on R whose membership function is given as follows: [X]i (y) =

max

(x1 ,...,xn )∈Rn xi =y

X(x1 , . . . , xn ).

(28)

It was proved in [23,27] that [X]i is a fuzzy number. It expresses the possibility distribution of the i-th variable (see e.g. [13]). In [23,28], it was shown that the joint possibility distribution of an n-tuple of uncertain weights can be appropriately modelled by an n-dimensional fuzzy vector W ∈ FV (Wn ). The possibility distributions of particular weights are in such a case given by the projections [W]1 , . . . , [W]n . An n-tuple of uncertain non-normalized weights is usually modelled by an n-tuple of non-interactive fuzzy weights Wi ∈ FN (R+ 0 ), Wi = {[w i (α), w i (α)]}α∈[0,1] , i = 1, . . . , n, such that w i (0) > 0 for at least one i ∈ {1, . . . , n}, i.e. at least one fuzzy weight is positive. Non-interactivity of the fuzzy weights means that there are no interactions among the values of weights (more details about non-interactivity of fuzzy numbers can be found e.g. in [7,10,13]). In such a case, the corresponding joint possibility distribution of the weights is expressed by the separable fuzzy vector of weights Wsep ∈ FV (Wn ) whose α-cuts are given as follows: Wsepα = W1α × · · · × Wnα ,

for all α ∈ (0, 1].

(29)

The fuzzy numbers W1 , . . . , Wn express the possibility distributions of particular weights, since Wi = [Wsep ]i , i = 1, . . . , n. However, by a fuzzy vector of weights, we can model also possible interactions among the non-normalized weights. For instance, the fuzzy vector of weights can be set in the following way: For n = 3, an expert describes uncertain values of weights by a triple of fuzzy numbers W1 , W2 , W3 ∈ FN (R+ ) and says that “the sum of the first two weights cannot be smaller than the third weight”. The fuzzy weights W1 , W2 , and W3 have to be “compatible” with the given interaction, there has to exist at least one weight vector (w1 , w2 , w3 ) ∈ W3 such that w1 + w2  w3 and wi ∈ Core Wi , i = 1, 2, 3. The joint possibility distribution of weights is then expressed by the 3-dimensional fuzzy vector of weights W whose α-cuts are for all α ∈ (0, 1] given as follows:    Wα = (w1 , w2 , w3 ) ∈ W3  wi ∈ Wiα , i = 1, 2, 3, w1 + w2  w3 . An n-tuple of uncertain normalized weights can be described by an n-dimensional fuzzy vector of normalized weights V ∈ FV (Sm ) (see [23,26,28]). Such a fuzzy vector expresses the joint possibility distribution of the n-tuple of normalized weights, and its projections [V]1 , . . . , [V]n to the particular axes express the possibility distributions of particular weights. Obviously, [V]i ∈ FN ([0, 1]) for any i ∈ {1, . . . , n}. As it was mentioned in Introduction, the most common way of modelling an n-tuple of uncertain normalized weights is to describe their values by fuzzy numbers V1 , . . . , Vn ∈ FN ([0, 1]) and to assume that only such combination of the weights are admissible in the model whose sum is equal to 1. Because of this interaction, a special structure of fuzzy numbers have to be applied for describing the values of normalized weights: We say that fuzzy numbers V1 , . . . , Vn ∈ FN ([0, 1]) form an n-tuple of normalized fuzzy weights, if, for all α ∈ (0, 1] and for all i ∈ {1, . . . , n}, the following condition is satisfied:

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For any vi ∈ Viα , there exist, for all j ∈ {1, . . . , n}, j = i, vj ∈ Vj α such that vi +

n 

vj = 1.

(30)

j =1, j =i

Let us note that the fuzzy numbers V1 , . . . , Vn ∈ FN ([0, 1]) satisfy the condition (30), if and only if their α-cuts satisfy the condition (7). In [23,26,28], it was shown that if an n-tuple of normalized fuzzy weights V1 , . . . , Vn represent the only available information about the values of normalized weights, the corresponding joint possibility distribution of an n-tuple of normalized weights is expressed by the fuzzy vector of normalized weights NV1 ,...,Vn ∈ FV (Sn ) whose α-cuts are for all α ∈ (0, 1] given as follows:    NV1 ,...,Vn α = (V1α × · · · × Vnα ) ∩ Sn = (v1 , . . . , vn ) ∈ Sn  vi ∈ Viα , i = 1, . . . , n . (31) Such a fuzzy vector of normalized weights was in [23,28] said to be separable on Sn . Furthermore, [NV1 ,...,Vn ]i = Vi , i = 1, . . . , n, i.e. V1 , . . . , Vn express the possibility distributions of the particular normalized weights. Finally, let us note that by a fuzzy vector of normalized weights V ∈ FV (Sn ), we can model also other kinds of interaction among the values of normalized weights than the fact that their sum is equal to 1. Such a fuzzy vector of normalized weights is not separable on Sn , i.e. it is not uniquely determined only by the n-tuple of normalized fuzzy weights and the set Sn . As we will see in next section, for n  3, this will be for instance the case of a fuzzy vector of normalized weights that will express the result of normalization of a fuzzy vector of non-normalized weights Wsep according to the extension principle. 4. Normalization of fuzzy vectors of weights according to the extension principle In this section, we will study the extension of the normalization procedure to the case when an n-tuple of uncertain initial non-normalized weights is modelled by a fuzzy vector of weights W ∈ FV (Wn ). The extension from functions having crisp arguments to functions with fuzzy set arguments is standardly done according to the extension principle proposed by Zadeh in [39]. 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(U ) → F(V ) such that for any A ∈ F(U ), the membership function of fF (A) is defined for all v ∈ V as follows:  supu∈U : f (u)=v A(u), if {u ∈ U | f (u) = v} = ∅, fF (A)(v) = (32) 0, otherwise. The fuzzy extensions of real-valued functions and real-vector-valued functions to general input fuzzy vectors were investigated in [23,27]. The normalization procedure is in the crisp case described by the real-vector-valued function n given by (1). Thus, according to the extension principle, the result of normalization of an initial fuzzy vector of weights W ∈ FV (Wn ) should be given by nF (W), where nF : F(Wn ) → F(Sn ) is the fuzzy extension of n given by (32). Since n is a continuous function, and since the membership function of any fuzzy vector of weights W ∈ FV (Wn ) is u.s.c., the α-cuts of nF (W) are for all α ∈ (0, 1] given as follows:  

  w1 wn  nF (W)α = (v1 , . . . , vn ) ∈ Sn (v1 , . . . , vn ) = n , . . . , n , (w1 , . . . , wn ) ∈ Wα . (33) i=1 wi i=1 wi The following theorem shows that the fuzzy extension nF has the property that the image nF (W) of any fuzzy vector of weights W ∈ FV (Wn ) is a fuzzy vector of normalized weights. Theorem 1. Let n : Wn → Sn be a real-vector-valued function defined for all w ∈ Wn by (1), and let nF be its fuzzy extension. If W ∈ FV (Wn ), then nF (W) ∈ FV (Sn ). Proof. The function n is continuous on Wn . Thus, according to [27, Theorem 24], it is sufficient to show that for all δ ∈ (0, 1) and for all x, y ∈ Wn , x = (x1 , . . . , xn ), y = (y1 , . . . , yn ), there exists λ ∈ [0, 1] such that   δ · n(x) + (1 − δ) · n(y) = n λ · x + (1 − λ) · y . This means that the following equations

O. Pavlaˇcka / Fuzzy Sets and Systems 243 (2014) 110–130

δ · xi (1 − δ) · yi λ · xi + (1 − λ) · yi n n + n = , λ · j =1 xj + (1 − λ) · nj=1 yj j =1 xj j =1 yj

121

i = 1, . . . , n,

must have the same solution λ ∈ [0, 1] independent of i. By solving the particular equations, we obtain δ · nj=1 yj λ = n , for all i ∈ {1, . . . , n}, δ · j =1 yj + (1 − δ) · nj=1 xj which completes the proof.

2

Remark 2. From (33), it can be easily seen that nF (V) = V for all V ∈ FV (Sn ). Thus, analogously as in the precise case, every fuzzy vector of normalized weights is normalized to itself. Remark 3. In the model, it may occur that the joint possibility distribution of an n-tuple of initial weights is expressed n by a fuzzy vector Wo ∈ FV ((R+ / 0 ) ) such that Supp Wo ⊂ Wn , but Cl(Supp Wo ) contains the zero vector, i.e. Wo ∈ FV (Wn ). For instance, such a fuzzy vector Wo can be obtained as the Cartesian product of fuzzy numbers Wi = {[wi (α), wi (α)]}α∈[0,1] , i = 1, . . . , n, where w i (0) = 0, i = 1, . . . , n, and, for at least one i ∈ {1, . . . , n}, wi (α) > 0 for all α ∈ (0, 1]. It is obvious that Core nF (Wo ) = ∅, that the α-cuts nF (Wo )α are for all α ∈ (0, 1] closed and convex, and that Supp nF (Wo ) is bounded (it is a subset of Sn ). Since also Cl(Supp nF (Wo )) ⊆ Sn , we obtain that nF (Wo ) ∈ FV (Sn ). n Particularly, if Wo ∈ FV ((R+ 0 ) ) is the Cartesian product of fuzzy numbers described above, and if w i (0) > 0 for all i ∈ {1, . . . , n}, i.e. if there are no zero weights, then Cl(Supp nF (Wo )) = Sn . In MCDM models, it is often convenient to compute the projections of the fuzzy vector of normalized weights nF (W) to the particular axes as they express the possibility distributions of particular normalized weights. We can proceed as follows: The way how the projections of an output of the fuzzy extension of a real-vector-valued function can be obtained in general is described in the following theorem that was proved in [23,27]. Theorem 4. Let D ⊆ Rm , m  1. Let f : D → Rn , n  2, be a real-vector-valued function with components fi : D → R, i ∈ Nn , i.e. f(x) = (f1 (x), . . . , fn (x)) for any x ∈ D. Let fiF , i = 1, . . . , n, denote the fuzzy extension of fi . Then for any X ∈ F(D), the following equalities hold:   fF (X) i = fiF (X), i = 1, . . . , n. (34) Let us apply the above theorem to the function n. We can write n = (n1 , . . . , nn ), where ni : Wn → [0, 1], i = 1, . . . , n, are defined for all (w1 , . . . , wn ) ∈ Wn in the following way: wi ni (w1 , . . . , wn ) := n . (35) j =1 wj Thus, according to (34), for any fuzzy vector of weights W ∈ FV (Wn ), it holds that   nF (W) i = niF (W), i = 1, . . . , n, i.e. the possibility distributions of particular normalized weights can be obtained by the fuzzy extension of the realvalued functions n1 , . . . , nn given by (35). Since these functions are continuous on Wn , we can employ the following well-known general result concerning the calculations with fuzzy vectors, the proof of which can be found e.g. in [34]. Theorem 5. Let D ⊆ Rm , m  1, let f : D → R be a continuous function, and let X ∈ FV (D). Let fF denote the fuzzy extension of f . Then fF (X) is a fuzzy number whose membership function is given by  maxx∈D: f (x)=y X(x), if {x ∈ D | f (x) = y} = ∅, fF (X)(y) = 0, otherwise. Furthermore,

   fF (X)α = f (Xα ) = y ∈ R  y = f (x), x ∈ Xα ,

for all α ∈ (0, 1].

(36)

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Remark 6. Employing the notation fF (X) = {[f X (α), f X (α)]}α∈[0,1] , it follows from (36) that for any α ∈ (0, 1], the values f X (α) and f X (α) can be obtained by solving the following two problems of mathematical programming: f X (α) = min f (x) x∈Xα

and

f X (α) = max f (x). x∈Xα

(37)

Furthermore, since the functions f X (α) and f X (α) are right-continuous at 0, the values f X (0) and f X (0) are given as follows: f X (0) = lim f X (α) α→0+

and f X (0) = lim f X (α). α→0+

(38)

According to Theorem 5, if W ∈ FV (Wn ), then niF (W) ∈ FN ([0, 1]) for all i ∈ {1, . . . , n}. Furthermore, let us denote niF (W) = {[niW (α), niW (α)]}α∈[0,1] , i = 1, . . . , n. Then, according to (37), the α-cuts of the projections of the fuzzy vector of normalized weights nF (W) to the particular axes are for all α ∈ (0, 1] given as follows: niW (α) = niW (α) =

min

wi n

,

(39)

max

wi n

.

(40)

(w1 ,...,wn )∈Wα

(w1 ,...,wn )∈Wα

j =1 wj j =1 wj

The following theorem shows that, for any fuzzy vector of weights W ∈ FV (Wn ), the fuzzy numbers n1F (W), . . . , nnF (W) that represent the projections of nF (W) to the particular axes form the n-tuple of normalized fuzzy weights. Theorem 7. Let n1 , . . . , nn be the functions given by (35) and n1F , . . . , nnF be their fuzzy extensions. Then, for any fuzzy vector of weights W ∈ FV (Wn ), the fuzzy numbers n1F (W), . . . , nnF (W) form the n-tuple of normalized fuzzy weights. Proof. Let us verify that the fuzzy numbers n1F (W), . . . , nnF (W) satisfy the condition (30). For any i ∈ {1, . . . , n} w∗ and α ∈ (0, 1], if vi ∈ niF (W)α , then it follows from (36) that there exist (w1∗ , . . . , wn∗ ) ∈ Wα such that vi = n i w∗ . k=1 k w∗ For j = 1, . . . , n, j = i, let us denote vj = n j w∗ . Then, obviously, vj ∈ nj F (W)α for all j , and vi + nj=1,j =i vj = k=1 k 1, which completes the proof. 2 A key role in computations with fuzzy vectors of normalized weights plays their potential separability on Sn . For instance, it was shown in [28] that if the fuzzy vector of normalized weights is separable on Sn , the computation of the fuzzy weighted average is significantly easier. Therefore, let us focus now on the problem whether the fuzzy vector of normalized weights nF (W) can be uniquely determined only by the n-tuple of the fuzzy numbers n1F (W), . . . , nnF (W) modelling the values of particular weights and by the relation Sn expressing all interactions among the weights, i.e. when nF (W) is separable on Sn . For n = 2, it has been shown in [28, Remark 7] that each 2-dimensional fuzzy vector of normalized weights is separable on S2 . Hence, for any W ∈ FV (W2 ),   nF (W)α = n1F (W)α × n2F (W)α ∩ S2 , for all α ∈ (0, 1]. (41) However, for n  3, the result of normalization of the initial fuzzy vector of weights W ∈ FV (Wn ) according to the extension principle cannot be expressed in general only by the n-tuple of normalized fuzzy weights n1F (W), . . . , nnF (W) and by the relation Sn . According to the general relation between any fuzzy vector and its projections proved in [23,27], only the following inclusion can be established for the α-cuts of nF (W):   nF (W)α ⊆ n1F (W)α × · · · × nnF (W)α ∩ Sn , for all α ∈ (0, 1]. (42) From (33), it follows that the potential separability of nF (W) on Sn depends on the initial fuzzy vector of weights W. The problem is illustrated by the following example.

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Example 3. For any c ∈ R+ , let 

 n   Qc := (w1 , . . . , wn ) ∈ Wn  wi = c . i=1

Let WQc ∈ FV (Qc ) be a fuzzy vector of weights such that   WQc α = [WQc ]1α × · · · × [WQc ]nα ∩ Qc . Let us show now that the fuzzy vector of normalized weights nF (WQc ) is separable on Sn . Let us denote:   [WQc ]i = w Qc i (α), wQc i (α) α∈[0,1] , i = 1, . . . , n, and 

   nF (WQc ) i = niWQc (α), niWQc (α) α∈[0,1] ,

i = 1, . . . , n.

Then, according to (39) and (40), for all i ∈ {1, . . . , n} and for any α ∈ (0, 1], niWQc (α) =

niWQc (α) =

min

wi n

=

max

wi n

=

wj ∈[w Qc j (α),w Qc j (α)] jn=1,...,n j =1 wj =c wj ∈[w Qc j (α),w Qc j (α)] jn=1,...,n j =1 wj =c

j =1 wj

j =1 wj

min

wi wQc i (α) = , c c

max

wi wQc i (α) = . c c

wi ∈[w Qc i (α),w Qc i (α)]

wi ∈[w Qc i (α),w Qc i (α)]

Further, the α-cuts of nF (WQc ) are for all α ∈ (0, 1] given as follows:    nF (WQc )α = n(w) ∈ Sn  w ∈ WQc α  

n     wn w1  n , . . . , n wi = c ∈ Sn  wi ∈ w Qc i (α), wQc i (α) , i = 1, . . . , n, = i=1 wi i=1 wi i=1      w Qc i (α) wQc i (α)  = (v1 , . . . , vn ) ∈ Sn  vi ∈ , , i = 1, . . . , n c c      = nF (WQc ) 1 × · · · × nF (WQc ) n ∩ Sn . Hence, nF (WQc ) is a fuzzy vector of normalized weights separable on Sn . 5. Properties of normalization of fuzzy vectors of weights according to the extension principle Now, let us focus on some important properties of the normalization procedure described in the previous section. The first important property consists in the fact that any α-cut of the resulting fuzzy vector of normalized weights nF (W) represents the set of all normalized weight vectors that are obtained by normalizing the weight vectors from Wα (it follows directly from (33)). Hence, nF (W) not only takes into account the condition of normalization that the sum of the normalized weights is equal to 1, but it also reflects all other interactions among the resulting normalized weights that follow from the initial fuzzy vector of weights W ∈ FV (Wn ). Similarly as the second desirable property of normalization of interval or fuzzy weights proposed by Sevastjanov et al. [32], the next important property of normalization of fuzzy vector of weights according to the extension principle is derived from the fact that in the precise case, the process of normalization preserves the ratios of initial nonzero weights. As it was mentioned in Section 2, Sevastjanov et al. [32] require the ratios of means of normalized intervals or fuzzy weights to be “as close as possible” to those of initial intervals or fuzzy weights. However, in our approach, we are dealing with fuzzy vectors expressing the joint possibility distributions of both the initial weight vector and the normalized weight vector. Hence, comparing the ratios of the means of fuzzy numbers expressing particular initial and normalized weights would not reflect the interactions between the weights. Moreover, comparing the fuzzy ratios

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[W]i [W]j

and

[nF (W)]i [nF (W)]j

computed according to the standard fuzzy arithmetic also would not be valid from the same reason. Thus, we have to focus on preserving the possibility distributions of the ratios of particular initial weights. They can be obtained in the following way: For all i, j ∈ {1, . . . , n}, let us denote the set of all n-dimensional weight vectors with positive i-th and j -th comij + ponents by Wn , i.e.    ij + Wn := (w1 , . . . , wn ) ∈ Wn  wi > 0, wj > 0 . (43) ij +

The ratio of the weight wi to the weight wj is in fact expressed by a real-valued function rij : Wn ij + all w ∈ Wn , w = (w1 , . . . , wn ), defined as follows: wi rij (w) := . wj

→ R+ that is for (44)

Let us note that the set of all n-dimensional vectors of normalized weights with positive i-th and j -th components is ij + a subset of Wn , i.e. the function rij expresses also the ratio of the i-th normalized weight to the j -th normalized weight. ij + Let rij F be the fuzzy extension of rij , and let W ∈ FV (Wn ) be a fuzzy vector of weights with positive i-th ij + and j -th components. Since rij is a continuous function on Wn , then, according to Theorem 5, rij F (W) is a fuzzy + number on R . For all α ∈ (0, 1], the α-cuts of rij F (W) are given as follows:   wi  rij F (W)α = r ∈ R+  r = , wi and wj express the i-th and j -th components wj  of at least one weight vector w ∈ Wα . Hence, it is clear that the fuzzy number rij F (W) expresses the possibility distribution of the ratio between the i-th and j -th weight, since it reflects all interactions between these two weights. The following theorem shows that the fuzzy extension nF preserves the possibility distributions of the ratios of particular initial weights. Theorem 8. Let nF be the fuzzy extension of the real-vector-valued function n defined by (1). Let for any i, j ∈ {1, . . . , n}, rij F be the fuzzy extension of the real-valued function rij given by (44). Then   ij +   (45) rij F (W) = rij F nF (W) , for all W ∈ FV Wn . Proof. Let us denote:   rij F (W) = r ij W (α), r ij W (α) α∈[0,1] , and     rij F nF (W) = r ij nF (W) (α), r ij nF (W) (α) α∈[0,1] . For all α ∈ (0, 1], r ij nF (W) (α) =

min

n(w)∈nF (W)α

    rij n(w) = min rij n(w) w∈Wα

(13)

= min rij (w) = r ij W (α). w∈Wα

The equality r ij W (0) = r ij nF (W) (0) follows from the right-continuity of r ij W and r ij nF (W) . The equalities r ij W (α) = r ij nF (W) (α) for all α ∈ [0, 1] can be derived analogously. 2 Now, let us focus on the third important property of the fuzzy extension of normalization of weights.

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The weights are in MCDM models mostly employed in the weighted average operation. The weighted average operation is a continuous real function a W : Wn × Rn → R defined for all w ∈ Wn , w = (w1 , . . . , wn ), and for all x ∈ Rn , x = (x1 , . . . , xn ), as follows: n i=1 wi · xi a W (w, x) := . (46) n i=1 wi The restriction of a W to Sn × Rn will be denoted by a N hereafter. For a normalized weight vector v ∈ Sn , v = (v1 , . . . , vn ), a N can be written in a simpler way: a N (v, x) :=

n 

v i · xi .

(47)

i=1

By normalization given by (1), we in fact transform any initial weight vector w ∈ Wn to the corresponding normalized weight vector n(w) ∈ Sn such that   a W (w, x) = a N n(w), x , for all x ∈ Rn . (48) Since the equality (48) plays a significant role in many applications, especially in MCDM models, its retaining also in the case of uncertain weights and weighted values would be very useful. In the following theorem, it is shown that the fuzzy extension nF satisfies the extension of the equality (48) to the case where both weights and weighted values are modelled by fuzzy vectors. Theorem 9. Let nF be the fuzzy extension of the real-vector-valued function n defined by (1). Let aFW and aFN be the fuzzy extensions of a W given by (46), and a N given by (47), respectively. Then, for all W ∈ FV (Wn ) and for all X ∈ FV (Rn ), the following equality holds:   aFW (W, X) = aFN nF (W), X . (49) W N N N Proof. Let aFW (W, X) = {[a W W,X (α), a W,X (α)]}α∈[0,1] , and aF (nF (W), X) = {[a nF (W),X (α), a nF (W),X (α)]}α∈[0,1] . For all α ∈ (0, 1],   (48) W N n(w), x aW W,X (α) = min a (w, x) = min a w∈Wα x∈Xα

=

min

n(w)∈nF (W)α x∈Xα

w∈Wα x∈Xα

  a N n(w), x = a N nF (W),X (α).

N W N The equality a W W,X (0) = a nF (W),X (0) follows from the right-continuity of a W,X and a nF (W),X . The equalities N aW W,X (α) = a nF (W),X (α) for all α ∈ [0, 1] can be derived analogously. 2

6. Normalization of non-interactive interval or fuzzy weights according to the extension principle Now, let us study normalization of initial non-interactive interval or fuzzy weights by the approach described in Section 4 and make a comparison of this approach with the methods of normalization of non-interactive interval and fuzzy weights described in Section 2. Since an n-tuple of interval weights [wi , wi ], i = 1, . . . , n, where 0  w i < wi for all i ∈ {1, . . . , n}, can be viewed as fuzzy weights Wi = {[wi (α), wi (α)]}α∈[0,1] , i = 1, . . . , n, of a special kind, where w i (α) = w i and wi (α) = wi for all α ∈ [0, 1], it is sufficient to consider further in this section only an n-tuple of non-interactive fuzzy weights. Let the initial non-normalized weights are modelled by an n-tuple of non-interactive fuzzy numbers Wi = {[wi (α), wi (α)]}α∈[0,1] , i = 1, . . . , n, such that wi (0)  0, i = 1, . . . , n, and wi (0) > 0 for at least one i ∈ {1, . . . , n}. According to Section 3, the corresponding joint possibility distribution of the initial weights is expressed by the fuzzy vector Wsep whose α-cuts are given by (29). The result of normalization of the fuzzy vector of weights Wsep according to the extension principle is the fuzzy vector of normalized weights nF (Wsep ) ∈ FV (Sn ) whose α-cuts are given for all α ∈ (0, 1] according to (33) as follows:

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  wi  nF (Wsep )α = (v1 , . . . , vn ) ∈ Sn  vi = n

j =1 wj

   , i = 1, . . . , n, wj ∈ wj (α), wj (α) , j = 1, . . . , n .

(50)

Let us note that in the case of non-normalized interval weights [w i , wi ], i = 1, . . . , n, the α-cuts nF (Wsep )α coincide with the set of normalized weight vectors V given by (10). For computing fuzzy numbers n1F (Wsep ), . . . , nnF (Wsep ) that express the possibility distributions of particular normalized weights, we can apply the formulas (37). Let us denote   niF (Wsep ) = niWsep (α), niWsep (α) α∈[0,1] , i = 1, . . . , n. Then, since the function ni given by (35) is on Wn non-decreasing with i-th variable and non-increasing with the rest, for all α ∈ [0, 1] and for any i ∈ {1, . . . , n}, niWsep (α) = niWsep (α) =

min

wj ∈[wj (α),w j (α)], j =1,...,n

max

wj ∈[wj (α),w j (α)], j =1,...,n

wi n

=

wi n

=

j =1 wj

j =1 wj

w i (α) + wi (α) +

w i (α) n

j =1, j =i

wi (α) n

j =1, j =i

wj (α)

wj (α)

,

(51)

.

(52)

Let us note that in the interval setting, these formulas coincide with the formulas (8) and (9) proposed by Wang and Elhag in [35]. It is worth to note that the fuzzy vector of normalized weights nF (Wsep ) is separable on Sn only if n = 2 (see the discussion in Section 4). For n  3, it was proved in [29] that if at least one initial fuzzy weight does not represent a real number, then each α-cut nF (Wsep )α is only a strict subset of the set   n1F (Wsep )α × · · · × nnF (Wsep )α ∩ Sn      = (v1 , . . . , vn ) ∈ Sn  vi ∈ niW (α), niW (α) , i = 1, . . . , n , (53) sep

sep

i.e. nF (Wsep ) is not separable on Sn . Now, let us focus on the problem of comparing the result of normalization of non-interactive interval or fuzzy weights according to the extension principle with the existing methods described in Section 2. Employing the methods of normalization proposed by Chang and Lee [3], Sevastjanov et al. [32], or Jiménez et al. [16] (see Section 2), we obtain only an n-tuple of non-interactive fuzzy numbers on [0, 1], not a fuzzy set on the set of all normalized weight vectors Sn . Moreover, since the resulting fuzzy numbers do not even form in general the n-tuple of normalized fuzzy weights (they may not fulfill the condition (30)), they cannot be interpreted as the possibility distributions of particular normalized weights. Therefore, it would be meaningless to compare the results of these methods with the result of normalization according to the extension principle directly (e.g. by a numerical example). The selection between these two different kinds of approaches should be based on the required interpretation of the fuzzy variables in the model. However, we can make a comparison with the method for normalization of non-interactive fuzzy weights proposed by Wang and Elhag [35]. The result of normalization of an initial n-tuple of non-interactive fuzzy weights is expressed by the n-tuple of normalized fuzzy weights n1F (Wsep ), . . . , nnF (Wsep ) (obtained by the formulas (51) and (52)) together with the constraint ni=1 vi = 1. According to Section 3, the joint possibility distribution that corresponds to this kind of information about the normalized weights is expressed by the fuzzy vector of normalized weights NWsep whose α-cuts are for all α ∈ (0, 1] given by (53). Hence, NWsep is separable on Sn . Let us compare now the fuzzy vectors of normalized weights nF (Wsep ) and NWsep that represent the results of both approaches to normalization of interval and fuzzy weights. The projections of both resulting fuzzy vectors of normalized weights to the particular axes are the fuzzy numbers n1F (Wsep ), . . . , nnF (Wsep ), i.e. the resulting possibility distributions of the particular normalized weights are the same. Let us focus on the resulting joint possibility distributions of the normalized weights. As it was mentioned earlier, each 2-dimensional fuzzy vector of normalized weights is separable on S2 . Hence, for n = 2, nF (Wsep ) = NWsep . However, for n  3, nF (Wsep )α ⊂ NWsep α ,

for all α ∈ (0, 1].

(54)

Since nF (Wsep )α represents for any α ∈ (0, 1] the set of all normalized weight vectors that are obtained by normalizing the weight vectors whose components belong to the α-cuts of the initial fuzzy weights W1 . . . , Wn , NWsep α contains

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also the normalized weight vectors that do not represent the result of normalization of the weights belonging to the α-cuts of initial fuzzy weights (see Example 1). Hence, NWsep does not express the correct joint possibility distribution of the resulting normalized weights. Furthermore, it immediately follows from (54) that only the following relationships can be established in general for the fuzzy vector of normalized weights NWsep : rij F (Wsep ) ⊆ rij F (NWsep ),

for any i, j ∈ {1, . . . , n},

(55)

  for all X ∈ FV Rn .

(56)

ij +

provided Wsep ∈ FV (Wn ), and aFW (Wsep , X) ⊆ aFN (NWsep , X),

Therefore, for n  3, it is not correct to express the result of normalization of non-interactive fuzzy weights W 1n, . . . , Wn only by the n-tuple of normalized fuzzy weights n1F (Wsep ), . . . , nnF (Wsep ) and the constraint i=1 vi = 1. It can cause a false increase of uncertainty in the model. The problem is illustrated by the following example. Example 4. Consider the following quadruple of triangular fuzzy weights which is taken from [35, Example 3]:   W1 = 0.08, 0.1, 0.3 = [0.08 + 0.02α, 0.3 − 0.2α] α∈[0,1] ,   W2 = 0.15, 0.2, 0.23 = [0.15 + 0.05α, 0.23 − 0.03α] α∈[0,1] ,   W3 = 0.2, 0.3, 0.6 = [0.2 + 0.1α, 0.6 − 0.3α] α∈[0,1] ,   W4 = 0.25, 0.4, 0.43 = [0.25 + 0.15α, 0.43 − 0.03α] α∈[0,1] . Let us assume that these fuzzy weights are provided independently and are not required to be summed to 1 (the first case in [35, Example 3]). The corresponding fuzzy vector expressing the joint possibility distribution of these initial fuzzy weights is the fuzzy vector Wsep whose α-cuts are given for all α ∈ (0, 1] by Wsep α = W1α × W2α × W3α × W4α . Let Vi = niF (Wsep ), i = 1, 2, 3, 4, be the quadruple of normalized fuzzy weights representing together with the constraint 4i=1 vi = 1 the result of normalization in [35, Example 3]). By NWsep , let us denote the fuzzy vector of normalized weights that expresses the corresponding joint possibility distribution of normalized weights. For all α ∈ (0, 1], the α-cuts of NWsep are given as follows: NWsep α = (V1α × V2α × V3α × V4α ) ∩ S4 . Further, let V = nF (Wsep ) be the fuzzy vector of normalized weights obtained by the extension principle that expresses the true joint possibility distribution of the resulting normalized weights. Although the possibility distributions of particular normalized weights coincide, i.e. [V]i = Vi for all i ∈ {1, 2, 3, 4}, according to (54), Vα ⊂ NWsep α holds for all α ∈ (0, 1]. First, let us show that the inclusion in (55) can be strict. For instance, let i = 1 and j = 2. The possibility distribution of the ratio of the first two initial weights is given by the fuzzy number RW , RW := r12F (Wsep ). According to Theorem 8, RW = r12F (V), i.e. RW also expresses the possibility distribution of the ratio of the first two normalized weights according to the extension principle. On the other hand, if the joint possibility distribution of the result of normalization is expressed by the fuzzy vector of normalized weights NWsep , the possibility distribution of the ratio of the first and second normalized weights is given by the fuzzy number RN , RN := r12F (NWsep ). The fuzzy numbers RW and RN are depicted in Fig. 1. We can easily see that RW ⊂ RN . Second, let us show that also the inclusion in (56) can be strict. Let the weighted values be, for simplicity, given by real numbers x1 = 7, x2 = 4, x3 = 10 and x4 = 1. The fuzzy weighted average of x1 , x2 , x3 and x4 with the fuzzy weights W1 , W2 , W3 and W4 is a fuzzy number XW , XW := aFW (Wsep , X) where X ∈ FV (R4 ) such that Xα = {(7, 4, 10, 1)} for all α ∈ (0, 1]. According to Theorem 9, XW = aFN (V, X). In contrast, the fuzzy weighted average of x1 , x2 , x3 and x4 with the normalized fuzzy weights V1 , V2 , V3 and V4 is a fuzzy number XN , XN := aFN (NWsep , X).

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Fig. 1. The fuzzy ratios RW and RN .

Fig. 2. The fuzzy weighted averages XW and XN .

The fuzzy numbers XW and XN are shown in Fig. 2. We can see that the fuzzy number XN is more uncertain than XW . 4Hence, considering the quadruple of normalized fuzzy weights V1 , V2 , V3 and V4 together with the constraint i=1 vi = 1 as the result of normalization of non-interactive fuzzy weights W1 , W2 , W3 and W4 , as it was in [35, Example 3], can cause a false increase of uncertainty in the model. 7. Conclusion In the paper, we have studied the problem of fuzzification of the procedure of normalization of weights. First, we have reviewed the existing methods for normalization of interval and fuzzy weights. The methods are based on different ways of understanding what the condition of normalization means in interval or fuzzy setting. We can distinguish two main approaches to modelling of uncertain normalized weights. In one approach, considered by Wang and Elhag in [35], the uncertain values of particular normalized weights are described by a special structure of intervals or fuzzy numbers on the unit interval and the restriction is considered further in the model that only such combination of the weights are admissible whose sum is equal to 1. Within this approach, we cannot apply standard interval or fuzzy arithmetic for computing with the normalized interval or fuzzy weights, a constraint interval or fuzzy arithmetic has to be used instead. The second approach, considered e.g. by Sevastjanov et al. in [32], consists in treating of normalized interval and fuzzy weights as interval and fuzzy objects, not as constraints on standard (real valued) normalized ones; no interaction among the normalized interval or fuzzy weights is considered further in the model. The selection between these two types of approaches should be based on the required interpretation of

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the interval and fuzzy variables in the model. Since the resulting normalized interval and fuzzy weights are in each method interpreted differently, they cannot be directly compared. Second, we have studied the problem of normalization of fuzzy vector of weights that express the joint possibility distribution of initial weights. We have shown that a correct way is to apply the extension principle proposed by Zadeh in [39] to the real-vector-valued function n given by (1). The result of such normalization is a fuzzy vector of normalized weights that expresses the joint possibility distribution of normalized weights; the possibility distributions of particular normalized weights are given as its projections. Further, we have proved some properties of this approach to normalization that are important from the point of view of real applications. Finally, since an n-tuple of non-interactive interval or fuzzy weights can be viewed as a fuzzy vector of weights of a special kind, we have investigated normalization of such kind of fuzzy vectors of weights according to the extension principle. We have shown that from the point of view of the way of modelling uncertain normalized weights, the result of this approach can be directly compared only with the result of the method proposed by Wang and Elhag in [35]. We have found out that for n = 2, both results coincide. However, in the case of n  3, we have shown that the result according to Wang and Elhag [35] does not express the correct joint possibility distribution of the normalized weights. This fact can cause a false increase of uncertainty in the model which has been illustrated by an example. 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