A new linguistic computational model based on discrete fuzzy numbers for computing with words

Information Sciences 258 (2014) 277–290

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A new linguistic computational model based on discrete fuzzy numbers for computing with words Sebastia Massanet a,⇑, Juan Vicente Riera a, Joan Torrens a, Enrique Herrera-Viedma b a b

Department of Mathematics and Computer Science, University of the Balearic Islands, Ctra. de Valldemossa, Km.7.5, 07122 Palma de Mallorca, Spain Department of Computer Science and Artificial Intelligence, University of Granada, Granada, Spain

a r t i c l e

i n f o

Article history: Received 19 January 2013 Received in revised form 30 April 2013 Accepted 22 June 2013 Available online 2 July 2013 Keywords: Discrete fuzzy number Subjective evaluation Multi-granular context Aggregation function

a b s t r a c t In recent years, several different linguistic computational models for dealing with linguistic information in processes of computing with words have been proposed. However, until now all of them rely on the special semantics of the linguistic terms, usually fuzzy numbers in the unit interval, and the linguistic aggregation operators are based on aggregation operators in [0, 1]. In this paper, a linguistic computational model based on discrete fuzzy numbers whose support is a subset of consecutive natural numbers is presented ensuring the accuracy and consistency of the model. In this framework, no underlying membership functions are needed and several aggregation operators defined on the set of all discrete fuzzy numbers are presented. These aggregation operators are constructed from aggregation operators defined on a finite chain in accordance with the granularity of the linguistic term set. Finally, an example of a multi-expert decision-making problem in a hierarchical multi-granular linguistic context is given to illustrate the applicability of the proposed method and its advantages. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction Uncertainty is a common factor in a wide range of real-world decision-making problems due to the tough task of handling it properly. This uncertainty often comes from the vagueness of meanings that are used by experts in problems where qualitative information is used. When data are qualitative, the fuzzy linguistic approach is a good tool to model them since these qualitative terms are represented via linguistic variables instead of numerical values (see [33–35]). These linguistic variables belong to a linguistic term set having more or less terms depending on the uncertainty degree, or the granularity of uncertainty, provided by the source of information qualifying an alternative. However, the use of the fuzzy linguistic approach have presented some limitations, specially regarding information modelling and computational processes, the so-called processes of computing with words (CW) (see [9,18,20]). To overcome these limitations, several different linguistic models have been presented: the symbolic linguistic model based on ordinal scales by assuming an ordered structure defined among the labels [31,14]; the linguistic 2-tuple model [16], which introduces the symbolic translation to the linguistic representation; the linguistic model based on type-2 fuzzy sets representation [27], which represents the semantics of the linguistic terms using type-2 membership functions and the proportional 2-tuple model [29], which extends the 2-tuple model by using two linguistic terms with their proportion to model the information, among many others.

⇑ Corresponding author. Tel.: +34 971259915. E-mail addresses: [email protected] (S. Massanet), [email protected] (J.V. Riera), [email protected] (J. Torrens), [email protected] (E. Herrera-Viedma). 0020-0255/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2013.06.055

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However, from an accurate analysis of these models, some common properties can emerge. First of all, there still exists some kind of limitation on the modelling of the linguistic information. The experts must express their evaluations on the alternatives in a single term that should be encompassed into a linguistic term set of a linguistic variable. This fact comes from that qualitative information is often interpreted to take values in a totally ordered finite scale like this:

L ¼ fExtremely Bad; Very Bad; Bad; Fair; Good; Very Good; Extremely Goodg: These linguistic terms will be denoted by EB, VB, B, F, G, VG and EG, respectively. In these cases, the representative finite chain Ln = {0, 1, . . . , n} is usually considered to model these linguistic hedges. However, the modelling of linguistic information is limited because the information provided by experts for each variable must be expressed by a simple linguistic term. In most cases, this is a problem because the expert’s opinion does not agree with a concrete term. On the contrary, experts’ values are usually expressions like ‘‘better than Good’’, ‘‘between Fair and Very Good’’ or even more complex expressions. In [25], using the hesitant fuzzy sets introduced by Torra in [26], a new model was proposed to increase the flexibility and richness of linguistic elicitation in hesitant situations under qualitative settings. The second common property of these models is the use of fuzzy numbers defined in the [0, 1] interval or their extensions to represent the semantics of the linguistic terms. The semantics are useful to capture the vagueness of the linguistic assessments given by the experts. The final remarkable common property is the use of the so-called linguistic aggregation functions (see [10,30]). These operators are used in order to merge the evaluations given by the experts into a representative output. These operators are very useful for modelling those processes in which there are various information sources and the information is linguistic in nature. However, most of these operators rely on an aggregation operator defined on the unit interval. First, the linguistic inputs are transformed into real numbers on the unit interval, the aggregation operator on the unit interval is applied and a transformation is performed in the output to recover a linguistic term. Another approach has been developed by several researchers through an extensive study of aggregation functions on Ln, usually called discrete aggregation functions (see [19,21,23]). Taking into account the previous considerations, we propose a linguistic computational model based on discrete fuzzy numbers whose support is a subset of consecutive natural numbers contained in Ln (i.e, an interval contained in Ln), usually denoted by AL1n (see [28] and [24]). The idea lies on the fact that any discrete fuzzy number A 2 AL1n can be considered (identifying the scale L given previously with n = 6) as an assignation of a [0,1]-value to each term in our linguistic scale. As an example, the above mentioned expression ‘‘between Fair and Very Good’’ can be performed, for instance, by a discrete fuzzy number A 2 AL16 , with support given by the subinterval [F, VG] (that corresponds to the interval [3, 5] in L6). The values of A in its support should be described by experts, allowing in this way a complete flexibility of the qualitative valuation overcoming the limitation on the modelling of the linguistic information. A possible discrete fuzzy number A representing the expression mentioned above is given in Fig. 1 (note that there are pictured only the values of A in its support). Thus, discrete fuzzy numbers can be interpreted as flexible qualitative information and they have already been successfully used in decision making problems and subjective evaluation in [24]. In addition, in this model, the semantics of the linguistic terms are included into the evaluation of the expert and there is no need of defining any underlying membership functions. On the other hand, the model is also useful to avoid the limitation of the aggregation functions on Ln or the internal use of aggregation functions into the linguistic aggregation functions. In [3,4,24], the authors deal with the possibility of extending monotonic operations on Ln to operations on the set of discrete fuzzy numbers whose support is a set of consecutive natural numbers AL1n . These aggregation functions on AL1n will allow us to manage qualitative information in a more flexible way and preserving all the model in a linguistic context. In [3] t-norms and t-conorms on AL1n are described and studied, as well as it is done for uninorms, nullnorms and general aggregation functions in [24]. In both cases, an example of application in decision making or subjective evaluation is included. This paper is structured as follows. In Section 2, we make a brief review of discrete fuzzy numbers and the linguistic hierarchical structure. In Section 3, the linguistic model based on discrete fuzzy numbers is proposed for a multiple expert decision making problem defined over a subjective linguistic hierarchy. Finally, in Section 4, we present an example of a multi-

Fig. 1. Graphical representation of a discrete fuzzy number whose support is the interval [2, 5]. In addition, note that this fuzzy set can be interpreted as expression ‘‘between Bad and Very Good’’, after identifying the linguistic scale L with the chain L6.

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expert decision-making problem in a hierarchical multi-granular linguistic context to illustrate the applicability of the proposed method and a discussion on its advantages and drawbacks. The paper ends with the conclusions and future work we want to develop. 2. Preliminaries In this section, we recall some definitions and results about aggregation functions and discrete fuzzy numbers which will be used later. We also recall the concept of a linguistic hierarchical structure. 2.1. Aggregation functions on bounded partially ordered sets Let (P; 6) be a non-trivial bounded partially ordered set (poset) with 0 and 1 as minimum and maximum elements respectively. Definition 1. An n-ary aggregation function on P is a function F: Pn ? P such that it is increasing in each component, F(0, . . . , 0) = 0 and F(1, . . . , 1) = 1.

Remark 1. Of course, the number of inputs to be aggregated can be different in each case. Thus, aggregation functions are commonly defined not on Pn, but on [nP 1 P n and then they are usually called extended aggregation functions. An easy way to construct extended aggregation functions is from associative binary aggregation functions. An important case is when we take as poset a finite chain Ln with n + 1 elements. In such a framework, only the number of elements is relevant (see [23]) and so the simplest finite chain, that is Ln = {0, 1, . . . , n}, is usually considered. On the other hand, it is well known that qualitative information is often interpreted to take values in a totally ordered finite scale L like this:

fExtremely Bad; Very Bad; Bad; Fair; Good; Very Good; Extremely Goodg:

ð1Þ

In these cases, the representative finite chain Ln is usually considered to model these linguistic hedges (L6 in previous scale (1)) and several researchers have developed an extensive study of aggregation functions on Ln, usually called discrete aggregation functions [7,8,19,21–23,32]. In Section 3.3, some examples of discrete aggregation functions will be used in order to generate aggregation functions defined on the set of discrete fuzzy numbers whose support is a set of consecutive natural numbers. 2.2. Discrete fuzzy numbers By a fuzzy subset of R, we mean a function A : R ! ½0; 1. For each fuzzy subset A, let Aa ¼ fx 2 R : AðxÞ P ag for any a 2 (0, 1] be its a-level set (or a-cut). By supp(A), we mean the support of A, i.e. the set fx 2 R : AðxÞ > 0g. By A0, we mean the closure of supp(A). Definition 2 [28]. A fuzzy subset A of R with membership mapping A : R ! ½0; 1 is called a discrete fuzzy number if its support is finite, i.e., there exist x1 ; . . . ; xn 2 R with x1 < x2 <    < xn such that supp(A) = {x1, . . . , xn}, and there are natural numbers s, t with 1 6 s 6 t 6 n such that:

1. A (xi) = 1 for any natural number i with s 6 i 6 t (core) 2. A(xi) 6 A(xj) for each natural number i, j with 1 6 i 6 j 6 s 3. A(xi) P A(xj) for each natural number i, j with t 6 i 6 j 6 n

Remark 2. If the fuzzy subset A is a discrete fuzzy number then the support of A coincides with its closure, i.e. supp(A) = A0. From now on, we will denote the set of discrete fuzzy numbers using the abbreviation DFN and dfn will denote a discrete fuzzy number. Also, we will denote by AL1n the set of discrete fuzzy numbers whose support is a subset of consecutive natural numbers contained in the finite chain Ln and by DLn the set of discrete fuzzy numbers whose support is a subset of natural numbers contained in Ln. Remark 3. Note that AL1n is a subset of DLn . Let A; B 2 AL1n be two discrete fuzzy numbers. Note that the supports of A and B are subintervals of Ln, and in fact, so is each h i   one of its a-cuts. Let Aa ¼ xa1 ; xap ; Ba ¼ ya1 ; yak be its a-level cuts for A and B, respectively. The following result holds for AL1n , but is not true for the set of discrete fuzzy numbers in general (see [2]).

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  Theorem 1 [2]. The triplet AL1n ; MIN; MAX is a bounded distributive lattice where 1n 2 AL1n (the unique discrete fuzzy number whose support is the singleton {n}) and 10 2 AL1n (the unique discrete fuzzy number whose support is the singleton {0}) are the maximum and the minimum, respectively, and where MIN(A, B) and MAX(A, B) are the discrete fuzzy numbers belonging to the set AL1n such that they have the sets

 n  o    and MINðA; BÞa ¼ z 2 Ln min xa1 ; ya1 6 z 6 min xap ; yak  n  o    MAXðA; BÞa ¼ z 2 Ln max xa1 ; ya1 6 z 6 max xap ; yak

ð2Þ

as a-cuts respectively for each a 2 [0, 1] and A; B 2 AL1n . Remark 4 [2]. Using these operations, we can define a partial order on AL1n in the usual way: A  B if and only if MIN(A, B) = A, or equivalently, A  B if and only if MAX(A, B) = B for any A; B 2 AL1n . Equivalently, we can also define the partial ordering in terms of a-cuts: A  B if and only if min (Aa, Ba) = Aa A  B if and only if max (Aa, Ba) = Ba

Remark 5. It is worth pointing out, as it has been already mentioned in the introduction, that the advantage of using discrete fuzzy numbers of the bounded partially ordered set AL1n lies on the fact that these can be interpreted as subjective evaluations made by experts (see Fig. 1). 2.3. Linguistic hierarchical structure The classical concept of linguistic hierarchy was introduced in [6] to design hierarchical systems of linguistic rules. Thus, a linguistic hierarchy [6,17] is a set of levels, where each level is a linguistic term set with different granularity to the rest of levels of the hierarchy. The granularity of uncertainty, i.e., the level of discrimination among different degrees of uncertainty is a relevant aspect to be analysed. Usually, odd values of cardinality such as 7 or 9 are used. In these linguistic term sets, there exists a mid term representing an assessment of approximately 0.5. In addition, the rest of the linguistic terms are placed symmetrically around it (see [1]). Each level belonging to a linguistic hierarchy is denoted as

lðt; nðtÞÞ being 1. t, a number that indicates the level of the hierarchy; 2. n(t), the granularity of the linguistic term set of the level t. In addition, we assume that the linguistic term sets have an odd value of granularity representing the central label the value of indifference. The levels belonging to a linguistic hierarchy are ordered according to their granularity, i.e., for two consecutive levels t and t + 1, n(t + 1) > n(t). This provides a linguistic refinement of the previous level. From the above concepts, a linguistic hierarchy, LH, is defined as the union of all levels t

LH ¼

[ lðt; nðtÞÞ: t

Generically, we can say that the linguistic term set of level t + 1 is obtained from its predecessor in the following way:

lðt; tðnÞÞ ! lðt þ 1; 2nðtÞ  1Þ as it can be observed in Table 1. The methodology is based in the two so-called linguistic hierarchy basic rules. The first one ensures the preservation of the former modal points of the membership functions of each linguistic term from one level to the following ones. The other one establishes a smooth transition between successive levels adding a new linguistic term between each pair of terms belonging to the term set of the previous level.

Table 1 Linguistic hierarchy. l (t, n (t)) Level 1 Level 2 Level 3

l (1, 3) l (2, 5) l (3, 9)

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Thus, as in [17], we start with S a linguistic term set over an universe of discourse U in the level t

S ¼ fs0 ; s1 ; . . . ; snðtÞ1 g being sk, with k = 0, . . . , n(t)  1, a linguistic term of S. Then in order to build a linguistic hierarchy, we extend the definition of S to a set of linguistic term sets Sn(t) where each term set belongs to a level t of hierarchy and has a granularity of uncertainty n(t)

n o nðtÞ nðtÞ SnðtÞ ¼ s0 ; . . . ; snðtÞ1 (see Fig. 2). Remark 6. We point out that in the literature the semantics of the linguistic terms are usually represented by fuzzy numbers whose membership functions are triangular or trapezoidal-shaped, symmetrical and uniformly distributed in [0, 1]. In addition, a unique membership function for each linguistic term is used throughout the decision-making process. However, as we will see later, we will consider a linguistic hierarchy where the semantics of each linguistic term is defined using a discrete fuzzy number belonging to AL1n that depends only on the expert, and which was not appointed previously. Thus, two experts can evaluate identical alternatives using different discrete fuzzy numbers that describe the same linguistic label. In this way the decision making process is more flexible for each evaluator because it allows a more personalized description of the linguistic labels.

3. Linguistic model based on discrete fuzzy numbers for decision problems defined over a subjective linguistic hierarchy In this section, we develop a new linguistic model based on the use of discrete fuzzy numbers in order to deal with multiexpert decision making problems where the evaluations of the experts are encompassed by a subjective linguistic hierarchy. 3.1. Subjective linguistic hierarchy The subjective linguistic hierarchy relaxes the well known concept of linguistic hierarchy since in this case, the linguistic term sets of each level do not require any semantics based on fuzzy numbers with some concrete membership functions. Definition 3. Let Ln = {0, . . . , n} be a finite chain. We call subjective evaluation to each discrete fuzzy number belonging to the partially ordered set AL1n . Remark 7. Note that we can consider a bijective mapping between the ordinal scale L ¼ fs0 ;    ; sn g and the finite chain Ln which keeps the original order. Furthermore, each normal continuous convex fuzzy subset defined on the ordinal scale L can be considered like a discrete fuzzy number belonging to AL1n , and vice versa. Thus, from now on a subjective evaluation A can be also interpreted equivalently as a normal continuous fuzzy set on the ordinal scale L. For example, consider the linguistic hedge

L ¼ fEB; VB; B; MB; F; MG; G; VG; EGg

ð3Þ

where the capital letters refer to the linguistic terms Extremely Bad, Very Bad, Bad, More or Less Bad, Fair, More or Less Good, Good, Very Good and Extremely Good and they are listed in an increasing order:

EB  VB  B  MB  F  MG  G  VG  EG and the finite chain L8. Thus, the subjective evaluation A = {0.6/MB, 0.7/F, 1/MG, 0.8/G} can be also expressed as A ¼ f0:6=3; 0:7=4; 1=5; 0:8=6g 2 AL18 (see Fig. 3). Once the notion of subjective evaluation has been presented, we define the concept of subjective linguistic hierarchy taking as semantics the subjective evaluations given by discrete fuzzy numbers in AL1n .

Fig. 2. Linguistic hierarchy of three, five and nine linguistic terms.

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Fig. 3. The triangles represent the subjective evaluation A.

Definition 4. A subjective linguistic hierarchy SLH is a linguistic hierarchy LH = are given by each subjective evaluation due to each expert.

S

tl(t, n(t))

were the semantics of the terms

Remark 8. Note that in the previous definition, the proposed semantics are associated with the subjective evaluation given by the expert but not with the concrete level of the chosen hierarchy as it happens in the classical linguistic hierarchies (see [6,17]). Thus, for instance, if we consider the previously defined linguistic hedge (3) the subjective evaluations A = {0.6/ MB, 0.7/F, 1/MG, 0.8/G} and B = {0.6/MB, 0.8/F, 1/MG, 0.9/G} are some possible evaluations made by two experts to the same alternative. 3.2. Completion functions among levels of a subjective linguistic hierarchy In multi-expert decision-making problems defined on multigranular linguistic terms sets, the experts give their valuations in linguistic scales with different granularity belonging to a linguistic hierarchy. For this reason, it is necessary to construct a transformation function among levels of a linguistic hierarchy (see for example [17,13]). In the case of a subjective linguistic hierarchy, these transformation functions will be based on the concept of completion of a discrete fuzzy number, which will be defined below. First, we introduce the concept of discrete association of a discrete fuzzy number. Definition 5. Let B 2 DLn be a discrete fuzzy number whose support is the set supp(B) = {x1, . . . , xv, . . . ,xp, . . . ,xm} with x1 <    < xv <    < xp <    < xm, m 6 n and B(xr) = 1 for all r such that v 6 r 6 p. A discrete association is a mapping

A:

DLn ! AL1n B # AðBÞ

such that for each discrete fuzzy number B 2 DLn , maps AðBÞ 2 AL1n fulfilling the following properties: 1. 2. 3. 4.

If xi 2 supp(B) then A(B)(xi) = B(xi) for each i = 1, . . . , m. B(xi) 6 A(B)(x) 6 B(xi+1), for all x 2 [xi, xi+1] with 1 6 i 6 i + 1 6 v. A(B)(xi) = 1, for all x 2 [xi, xi+1] with v 6 i 6 i + 1 6 p. B(xi+1) 6 A(B)(xi) 6 B(xi), for all x 2 [xi, xi+1] with p 6 i 6 i + 1 6 m.

Discrete associations allow to construct from a discrete fuzzy number whose support is not a subset of consecutive natural numbers, i.e., a dfn in DLn , a discrete fuzzy number whose support is a subset of consecutive natural numbers, i.e., a dfn in AL1n . The association preserves the values in the support of the original dfn and it assigns evaluations to those natural numbers of Ln with no previous evaluation in a way which preserves the properties of a dfn. S L 1 Definition 6. Let us consider a subjective linguistic hierarchy SLH = t l(t, n(t)). Let A 2 A1nðtÞ be a subjective evaluation n o nðtÞ nðtÞ nðtÞ nðtÞ nðtÞ nðtÞ nðtÞ nðtÞ with s0 <    < sv <    < sp <    < sm and whose support is the set suppðAÞ ¼ s0 ; . . . ; sv ; . . . ; sp ; . . . ; sm   nðtÞ ¼ 1 for all r such that v 6 r 6 p and m 6 n(t)  1 6 n(k)  1. A completion is a mapping A sr

C:

L

L

A1nðtÞ1 ! A1nðkÞ1 A # AðBÞ

    nðkÞ nðtÞ for each i = 0, . . . ,n(t)  1. where B 2 DLnðkÞ1 such that B s2kt i ¼ A si

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According to the previous definition, given some A 2 A1nðtÞ1 , i.e., a dfn belonging to a level of the subjective linguistic hierarchy with n(t) linguistic terms, a completion to another level n(k) > n(t) can be split in two steps: nðkÞ

nðtÞ

1. We assign A to B 2 DLnðkÞ1 assigning to s2kt i the evaluation of the linguistic term si . 2. We use a discrete association in order to obtain from B 2 DLn ðkÞ a dfn with support a subset of consecutive natural numbers. After these two steps, from a subjective evaluation given in a linguistic term set at some level n(t), we obtain a subjective evaluation given in a linguistic term set at a higher level n(k) containing the information of the original evaluation. Now we are going to give several examples of completions. L

L

Example 1. An a-completion, Ca , is a completion such that for each A 2 A1nðtÞ1 maps Ca ðAÞ 2 A1nðkÞ1 fulfilling:

h i 8  nðtÞ  nðkÞ nðkÞ nðtÞ nðtÞ > A si if x 2 s2kt i ; s2kt ðiþ1Þ with siþ1 6 sv ; > > > < h i nðkÞ nðkÞ Ca ðAÞðxÞ ¼ 1 if x 2 s2kt v ; s2kt p ; > >   h i > > nðkÞ nðkÞ nðtÞ nðtÞ : A snðtÞ if x 2 s2kt i ; s2kt ðiþ1Þ with si P sp : iþ1 For instance: L Let A ¼ f0:4=0; 1=1; 0:8=2g 2 AL12 ¼ A1nð1Þ1 be, then

8 > < 0:4 if x 2 f0; 1; 2; 3g L Ca ðAÞðxÞ ¼ 1 2 A1nð3Þ1 if x 2 f4g > : 0:8 if x 2 f5; 6; 7; 8g The next Fig. 4 shows the transformation process of the discrete fuzzy number A using the a-completion defined above. L If we get A ¼ f0:5=0; 1=1; 0:7=2g 2 AL12 ¼ A1nð1Þ1 be, then

8 > < 0:5 if x 2 f0; 1g L L Ca ðAÞðxÞ ¼ 1 if x 2 f2g 2 A14 ¼ A1nð2Þ1 > : 0:7 if x 2 f3; 4g L

L

Example 2. An x-completion, Cx , is a completion such that A 2 A1nðtÞ1 maps Cb ðAÞ 2 A1nðkÞ1 defined as

8   nðkÞ > > if x ¼ s2kt j for any 1 6 j 6 m A snj ðtÞ > > > >   >   > nðkÞ nðkÞ nðtÞ nðtÞ > < A sniþ1 ðtÞ if x 2 s2kt i ; s2kt ðiþ1Þ with siþ1 6 sv Cb ðAÞðxÞ ¼   nðkÞ nðkÞ > > 1 if x 2 s2kt v ; s2kt p > > > >     > > nðkÞ nðkÞ nðtÞ nðtÞ > : A sinðtÞ if x 2 s2kt i ; s2kt ðiþ1Þ with si P sp

L

Fig. 4. The triangles represent the initial points of A 2 A1nð1Þ1 ¼ AL12 , while the circles correspond to the added points.

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For instance: L Let A ¼ f0:8=1; 1=2; 1=3; 0:6=4g 2 AL14 ¼ A1nð2Þ1 be, then

8 < 0:8 if x 2 f2; 3g L L Cb ðAÞðxÞ ¼ 1 if x 2 f4; 5; 6g 2 A18 ¼ A1nð3Þ1 : 0:6 if x 2 f7; 8g

Next Fig. 5 depicts the transformation process of the x-completion. Remark 9. Note that while the a-completion assigns to a linguistic term without any original evaluation the minimum of the evaluations surrounding it, the x-completion assigns it the maximum. In fact, any function F such that min 6 F 6 max could be used as a completion function. The completion concept defined previously allows us a normalization process, that is, to represent in a common linguistic scale the different valuations made by experts corresponding to a concrete alternative. Next step in the decision making process is to find an aggregation procedure of these normalized valuations (see [11]). 3.3. Aggregation of subjective evaluations In order to merge subjective evaluations it is necessary to construct appropriate aggregation functions which combine these valuations. As result of such fusion we obtain another subjective evaluation which represents the collective preference. In this way, the authors (see [3,4,24]) deal with the construction of aggregation functions defined on the set of discrete fuzzy numbers whose support is a subset of consecutive natural numbers contained in a finite chain Ln and the particular cases of t-norms, t-conorms, uninorms and nullnorms are studied in detail. These aggregation functions are constructed from discrete aggregation functions (defined on a finite chain) and they will be applied to the aggregation of subjective evaluations. The main result is as follows: Theorem 2 ([3,24]). Let us consider a binary aggregation function F on the finite chain Ln. The binary operation on AL1n defined as follows

F : AL1n  AL1n ! AL1n ðA; BÞ # F ðA; BÞ being F ðA; BÞ the discrete fuzzy number whose a-cuts are the sets

fz 2 Ln j min FðAa ; Ba Þ 6 z 6 max FðAa ; Ba Þg for each a 2 [0, 1] is an aggregation function on AL1n . This function will be called the extension of the discrete aggregation function F to AL1n . In particular, if F is a t-norm, a t-conorm, an uninorm or a nullnorm its extension F , too. In addition, if F is a compensatory aggregation function in Ln, its extension F , too. Example 3. Let us consider B ¼ f0:3=2; 0:5=3; 1=4; 0:8=5g 2 AL18 .

the

finite

chain

L8 = {0,1,2,3,4,5,6,7,8}

and

A ¼ f0:3=0; 0:5=1; 1=2; 0:3=3g;

 Consider the discrete idempotent uninorm [7]

Uðx; yÞ ¼



minðx; yÞ if y 6 8  x maxðx; yÞ otherwise

L

Fig. 5. The triangles represent the initial points of A 2 A1nð1Þ1 ¼ AL14 , while the circles correspond to the added points.

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defined on the finite chain L8. Then, UðA; BÞ ¼ f0:3=0; 0:5=1; 1=2; 0:3=3g. According to Remark 7, A and B can be rewritten like

A ¼ f0:3=EB; 0:5=VB; 1=B; 0:3=MBg B ¼ f0:3=B; 0:5=MB; 1=F; 0:8=MGg: So, UðA; BÞ ¼ f0:3=EB; 0:5=VB; 1=B; 0:3=MBg.  Consider the discrete nullnorm defined on L8

8 if ðx; yÞ 2 ½0; 42 > < minðx þ y; 4Þ Fðx; yÞ ¼ maxð4; x þ y  8Þ if ðx; yÞ 2 ½4; 82 > : 4 otherwise Then F ðA; BÞ ¼ f0:3=B; 0:3=MB; 1=Fg. However, the previous classes of aggregation functions have a drawback. T-norms, t-conorms, nullnorms and uninorms are not compensatory aggregation functions in their whole domain. However, in [22], a compensatory class of aggregation functions on the finite chain Ln given by

Fðx1 ; . . . ; xm Þ ¼ maxfminfx1 ; . . . ; xm g; maxfx1 ; . . . ; xm g  kg for some k 2 [0, n  1] was presented. Using these functions, we can obtain compensatory aggregation functions on the set of dfn’s in AL1n in the sense that the output is between MIN and MAX (as defined in Theorem 1) of the inputs. Example 4. In the same conditions of Example 3, if we consider F(x, y) = max{min{x, y},max{x, y}  2}, we obtain F ðA; BÞ ¼ f0:3=EB; 0:5=VB; 1=B; 0:8=MBg. Remark 10. In the literature, one of the most important linguistic aggregation operators are the OWA operators and their generalizations. In our framework, it is also possible to define OWA type aggregation functions from OWA functions defined on Ln. However, in this discrete chain, these functions need a rounding function in order to get as output a value of the chain (see [19]). This fact could produce a loss of accuracy throughout the process and thus, the previous discrete aggregation functions are preferable since all the computations always remain into the domain of the finite chain. 3.4. Exploitation of collective evaluations In previous sections we dealt with the aggregation phase in order to obtain a collective preference value for each possible alternative based on the subjective evaluations given by the experts. Now, we need to choose the best one, i.e., to exploit the collective linguistic preference (see [11]). For this reason we use the ranking method proposed by L. Chen and H. Lu in [5]. Let us summarize this procedure briefly. For each discrete fuzzy number A, the lower and upper limits of the kth a-cut for A are defined as

li;k ¼ minfxjAðxÞ P ak g

ð4Þ

ri;k ¼ maxfxjAðxÞ P ak g

ð5Þ

x2R

x2R

respectively, where li,k and ri,k are the left and right spreads, respectively. The left (right) dominance DLi;j ðDRi;j Þ of Ai over Aj is defined as the average difference of the left (right) spreads at some alevels. They are formulated as

DLi;j ¼

n 1 X ðli;k  lj;k Þ n þ 1 k¼0

ð6Þ

DRi;j ¼

n 1 X ðri;k  r j;k Þ n þ 1 k¼0

ð7Þ

and

were n + 1 a-cuts are used to calculate the dominance. The total dominance of Ai over Aj with the index of optimism b 2 [0, 1] can be defined as the convex combination of DLi;j and DRi;j by

"

Di;j ðbÞ ¼

bDRi;j

þ ð1 

bÞDLi;j

# " # n n 1 X 1 X ¼b ðr i;k  r j;k Þ þ ð1  bÞ ðli;k  lj;k Þ n þ 1 k¼0 n þ 1 k¼0

ð8Þ

The above equation indicates that the total dominance is actually a comparison function. The larger the index of optimism b is, more important is the right dominance. Herein, the index of optimism is used to reflect a decision maker’s degree of optimism. A more optimistic decision maker generally takes a larger value of the index, for example, a situation in which b = 1 (or 0) represents an optimistic (pessimistic) decision maker’s perspective, and only right (left) dominance is considered.

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Furthermore, the total dominance of one fuzzy number over the other equals the difference between the two convex combinations using the respective left and right spreads. A decision maker can rank a pair of fuzzy numbers, Ai and Aj, using Di,j(b) according to the following rules: 1. If Di,j(b) > 0 then Ai > Aj; 2. If Di,j(b) = 0 then Ai = Aj; 3. If Di,j(b) < 0 then Ai < Aj. This distance between discrete fuzzy numbers satisfies some desirable properties, collected in the next result. Proposition 3 [5]. Let Ai, Aj and Ak be any three arbitrary fuzzy numbers, where i – j – k. Then: 1. The total dominance of a dfn over itself is 0, i.e., Di,i(b) = 0 for any i and b. 2. The total dominance of Ai over Aj is opposite to that of Aj over Ai, i.e., Di,j(b) =  Di,j(b) for any i, j and b. 3. The transitivity property holds: if Di,j(b) > 0 and Dj,k(b) > 0, then Di,k(b) > 0.

Example 5 1. Let us consider the finite chain L8 = {0, 1, 2, 3, 4, 5, 6, 7, 8}, A1 = {0.1/2, 0.8/3, 1/4, 0.8/5} and A2 ¼ f0:7=3; 1=4; 0:1=5g 2 AL18 . In this case, it holds that DL1;2 ¼ DR1;2 and D1,2(b) = b  0.5. So, when b < 0.5, D1,2(b) < 0 and we promote the left dominance of A2 over A1, while when b > 0.5, D1,2(b) > 0 and we enhance the right dominance of A1 over A2. 2. Taking again L8, let us consider A1 ¼ f0:3=2; 0:5=3; 1=4; 0:8=5g; A2 ¼ f0:5=2; 1=3; 0:9=4g 2 AL18 . Then an easy computation shows that D1,2(b) = 0.8 > 0 for all b 2 [0,1] showing that A1 dominates A2, i.e., A1 > A2. 3. Consider again L8 but now A1 ¼ f0:6=2; 1=3; 0:7=4; 0:2=5g; A2 ¼ f0:8=2; 1=3; 0:9=4g 2 AL18 . In this case, we have that D1,2(0.6) = 0.033 > 0 and consequently, A1 > A2 with b = 0.6, but D1,2(0.7) =  0.0167 < 0 and A2 > A1 with b = 0.7. As b increases, the right dominance of A2 with respect to A1 overcomes the left dominance of A1 with respect to A2. 4. Example In this section, we develop a multi-expert decision making problem defined in a multi-granular linguistic context as an application of the model presented in the previous section. The example is similar to the one developed in [17]. In these real-world situations the experts express their opinions by means of linguistic terms belonging to different scales depending on the different granularity of uncertainty they want to manage, i.e., they use multi-granular linguistic information (see [12]). The only assumption is that the different scales used S by the experts are into the subjective linguistic hierarchy SLH = tl(t, n(t)) with t = 1, 2, 3. 4.1. Description For the sake of completeness, we will recall the example. Let us suppose an investment company, which wants to invest a sum of money in the best option. There is a panel with four possible alternatives to invest the money:    

x1 x2 x3 x4

is is is is

a a a a

car industry, computer company, food company, weapon industry.

The investment company has a group of four consultancy departments:    

e1 e2 e3 e4

is is is is

the the the the

risk analysis department, growth analysis department, social-political analysis department, environmental impact analysis department.

Each department is directed by an expert, and thus, each expert is an information source. These experts use to provide their preferences, over the set of alternatives, from different term sets of the linguistic hierarchy depending of the degree of uncertainty they desire. Specifically,  e1 provides his preferences in l(3, 9),  e2 provides his preferences in l(2, 5),

S. Massanet et al. / Information Sciences 258 (2014) 277–290

287

Table 2 Expert group sheet for alternatives x1 and x2. Expert

Alternative x1

Alternative x2

e1 e2 e3 e4

{0.5/2, 0.7/3, 1/4, 0.8/5, 0.7/6} {0.7/2, 1/3, 0.8/4} {1/1, 0.9/2} {0.5/3, 1/4, 0.8/5, 0.7/6, 0.7/7}

{0.8/4, 0.9/5, 1/6, 0.7/7, 0.6/8} {0.7/2, 0.8/3, 1/4} {0.3/0, 0.5/1, 1/2} {0.7/3, 0.8/4, 1/5, 0.8/6, 0.7/7}

Table 3 Expert group sheet for alternatives x3 and x4. Expert

Alternative x3

Alternative x4

e1 e2 e3 e4

{0.6/1, 0.7/2, 1/3, 0.8/4} {0.5/2, 1/3, 0.4/4} {0.1/0, 0.1/1, 1/2} {0.8/0, 0.9/1, 0.9/2, 1/3, 0.5/4}

{0.7/3, 0.8/4, 1/5, 0.7/6} {0.1/1, 0.2/2, 1/3, 0.6/4} {0.9/0, 1/1, 0.1/2} {0.8/4, 1/5, 0.9/6}

 e3 provides his preferences in l(1, 3),  e4 provides his preferences in l(3, 9). After a deep analysis, the experts provide the subjective evaluations for each alternative (see Tables 2 and 3). We will denote by Oji the subjective evaluation of alternative xj given by expert ei for i, j = 1, . . . , 4. Remark 11. For the sake of an easier understanding of the process, we have expressed the linguistic terms by means of the elements of Ln using the bijection between them. Thus, for instance, we have that O11 ¼ f0:5=B; 0:7=MB; 1=F; 0:8=MG; 0:7=Gg. Remark 12. In order to make a fair comparison with the 2-tuple model presented in [17], the core of every subjective evaluation is located at the linguistic label chosen by the expert as its judgment in the example presented in [17]. Note that subjective evaluations allow a more flexible way to evaluate an alternative since every linguistic label of the chain has a particular valuation according to the confidence of the expert. 4.2. Decision model The decision model used to solve the problem considered in this section is based on two steps (see [11]): an aggregation phase and an exploitation phase. 4.2.1. Aggregation phase In this step, the information must be combined to obtain a collective preference value for each alternative. However, since the experts have chosen different levels of the subjective linguistic hierarchy in order to give their evaluations for the alternatives, a normalization process must be carried out. In this normalization step, a linguistic term set of the subjective linguistic hierarchy is chosen to make uniform the multigranular linguistic information. Since we have two subjective evaluations in l(3, 9), every other alternative must be expressed in terms of the linguistic labels of this level of the hierarchy. This is achieved through the use of a completion function. In this example, we have used the a-completion. Thus, the normalized evaluations are given in Tables 4 and 5. b j the subjective evaluation of alternative xj given by expert ei for i, j = 1, . . . , 4. We will denote by O i Once the normalized subjective evaluations are obtained, an aggregation function on AL18 is applied to obtain the collective subjective evaluation on each alternative. In this example, we have used the extension F to the set of AL18 of the compensatory discrete aggregation function

Fðx1 ; x2 ; x3 ; x4 Þ ¼ maxfminfx1 ; x2 ; x3 ; x4 g; maxfx1 ; x2 ; x3 ; x4 g  4g;

Table 4 Normalized expert group sheet for alternatives x1 and x2. Expert

Alternative x1

Alternative x2

e1 e2 e3 e4

{0.5/2, 0.7/3, 1/4, 0.8/5, 0.7/6} {0.7/4, 0.7/5, 1/6, 0.8/7, 0.8/8} {1/4, 0.9/5, 0.9/6, 0.9/7, 0.9/8} {0.5/3, 1/4, 0.8/5, 0.7/6, 0.7/7}

{0.8/4, 0.9/5, 1/6, 0.7/7, 0.6/8} {0.7/4, 0.7/5, 0.8/6, 0.8/7, 1/8} {0.3/0, 0.3/1, 0.3/2, 0.3/3, 0.5/4, 0.5/5, 0.5/6, 0.5/7, 1/8} {0.7/3, 0.8/4, 1/5, 0.8/6, 0.7/7}

288

S. Massanet et al. / Information Sciences 258 (2014) 277–290 Table 5 b 3 ¼ f0:1=0; 0:1=1; 0:1=2; 0:1=3; 0:1=4; 0:1=5; 0:1=6; 0:1=7; 1=8g Normalized expert group sheet for alternatives x3 and x4 where O 3 b 4 ¼ f0:9=0; 0:9=1; 0:9=2; 0:9=3; 1=4; 0:1=5; 0:1=6; 0:1=7; 0:1=8g. and O 3 Expert

Alternative x3

Alternative x4

e1 e2 e3

{0.6/1, 0.7/2, 1/3, 0.8/4} {0.5/4, 0.5/5, 1/6, 0.4/7, 0.4/8} b3 O

{0.7/3, 0.8/4, 1/5, 0.7/6} {0.1/2, 0.1/3, 0.2/4, 0.2/5, 1/6, 0.6/7, 0.6/8} b4 O

e4

{0.8/0, 0.9/1, 0.9/2, 1/3, 0.5/4}

{0.8/4, 1/5, 0.9/6}

3

3

Table 6 Collective subjective evaluations obtained using F . Alternative

Collective subjective evaluation

x1 x2 x3 x4

{0.5/2, 0.7/3, 1/4, 0.8/5, 0.7/6} {0.3/0, 0.3/1, 0.3/2, 0.5/3, 0.8/4, 1/5, 0.8/6, 0.7/7} {0.1/0, 0.1/1, 0.1/2, 0.1/3, 1/4} {0.2/0, 0.2/1, 0.9/2, 0.9/3, 1/4, 0.1/5, 0.1/6}

which is an example of a discrete kernel aggregation function studied recently in [22]. The collective subjective evaluations are the following ones (see Table 6): We will denote by Ai the collective subjective evaluation of alternative xi for i = 1, . . . , 4. These collective subjective evaluations are displayed in Fig. 6. 4.2.2. Exploitation phase In this phase, we want to choose the alternative with the best collective subjective evaluation. For this purpose, we will use the dominance distance presented in Section 3.4 with b = 0.7. We have chosen this value for b since we want to strengthen the right dominance over the left dominance. However, depending on the judgement of the decision maker, any other value b 2 (0, 1) could be also considered. In this case, an easy computation shows that D2,1 = 0.76 > 0, D1,3 = 0.92 > 0 and D3,4 = 0.1 > 0 and due to the transitivity property of this distance, we can conclude that

x2 > x1 > x3 > x4 : Thus, the best option to invest the money is x2. Remark 13. Note that our model and the 2-tuple model (see [17]) agree with the best and worst alternative. On the other hand, the model based on discrete fuzzy numbers appoints x1, the car industry as a better option than x3, the food company while the 2-tuple model exchange them. However, in the 2-tuple model both alternatives are notably close since x3 gets a     collective preference value of s95 ; 0:25 while x1 gets s95 ; 0:5 , so the gap is quite small. 4.3. Discussion In this subsection we highlight some advantages of our model with respect to others existing in the literature and point out some aspects that should be improved too. We focus on two aspects to analyse the advantages of our proposal: 1. Complexity of the linguistic representation model: As aforementioned, all important fuzzy linguistic approaches existing in the literature such as the symbolic linguistic model [14,31], the linguistic 2-tuple model [16], the linguistic model based on type-2 fuzzy linguistic approach [27] and the proportional 2-tuple fuzzy linguistic approach [29] use fuzzy numbers defined in the [0, 1] interval or their extensions to represent the semantics of the linguistic terms. As it is known this is a sensible aspect in the definition of linguistic domains whose difficulty is high because we have to set the parameters of the membership functions associated with labels and experts may have different views on the interpretation of the labels, and the agreement between them is not always easy [11,15]. This problem is overcome with our approach because it is independent of the membership functions associated with labels. 2. Complexity of the computational model to aggregate linguistic information: The symbolic linguistic model [14,31] presents limitations in the aggregation processes of linguistic information due to the application of rounding operations. The linguistic 2-tuple model [16] and the proportional 2-tuple fuzzy linguistic approach [29] overcome that drawback and allow us to define aggregation operators to combine linguistic information without loss of information. However, they apply internal procedures based on different transformation functions that increase the complexity of the computational models. With our linguistic approach we can use the well defined mathematical framework of discrete fuzzy numbers for defining similar aggregation operators of linguistic information but avoiding the limitation of the internal use of transformation functions.

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289

Fig. 6. Collective subjective evaluations of the alternatives x1, x2, x3 and x4, respectively.

On the other hand, as in [25] our fuzzy linguistic approach increases the flexibility and richness of linguistic elicitation, but we find that sometimes for some experts could be difficult to use our linguistic representation model based on subjective assessment to express their preferences, and therefore we have to work to design appropriate interfaces and procedures to aid users to complete our preference representation format. 5. Conclusions In this paper, we have proposed a linguistic computational model based on the use of discrete fuzzy numbers whose support is a subset of consecutive natural numbers of a finite chain Ln. This model allows a greater flexibility on the evaluations given by the experts, who can evaluate in a soft way giving a mark to the linguistic labels they consider appropriate. The model ensures the accuracy and consistency of the process since no approximation or rounding operation is needed. In addition, the model avoids the use of any continuous operation or membership function in the unit interval restricting itself to closed operations in the set of discrete fuzzy numbers. Finally, we have presented an application of this model to a realworld problem where the experts evaluate a set of alternatives with linguistic term sets belonging to a subjective linguistic hierarchy.

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As future work, we want to generalize the model to deal with problems where the experts could evaluate the alternatives in different linguistic term sets without the restriction of belonging to a linguistic hierarchy and also by means of unbalanced linguistic term sets [13]. In addition, the compensatory operators on the set of discrete fuzzy numbers are worth to study in order to have a wider range of options in the aggregation phase. Acknowledgments This paper has been partially supported by the Spanish Grants MTM2009-10320, MTM2009-10962 and also with the financing of FEDER funds in FUZZYLING-II Project TIN2010-17876, Andalusian Excellence Projects TIC-05299 and TIC-5991. References [1] P.P. Bonissone, K.S. Decker, Selecting Uncertainty Calculi and Granularity: An Experiment in Trading-Off Precision and Complexity, vol. 221, NorthHolland, Amsterdam, The Netherlands, 1986. [2] J. Casasnovas, J. 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