(classical and ordinal(Herrera Herrera-Viedma, 1997: Zadeh, that transform without loss of information numerical values 1975)). To define it we have to establish the 2-tuple representation into linguistic 2-tuples and viceversa, any of the existing model and the 2-tuple computational model to represent and aggregation operator can be easily extended for dealing with aggregate the linguistic information, respectively linguistic 2-tuples. Some examples are: Let s= Sgt be a linguistic term set with odd cardinali where the mid term represents a indifference value and the rest of Definition 3(Arithmetic mean). Let x=[( 1, 1),.(rn, Mn))be a the terms is symmetrically related to it. We assume that the set of linguistic 2-tuples, the 2-tuple arithmetic mean X is semantics of the labels is given by means of triangular membership computed as, strib 0nwha0edns1 In this fuzzy lin- X[,2)…(m=4(∑(,x)=4 guistic context, if a symbolic method(Herrera Herrera-Viedma 1997: Herrera et al., 1996) aggregating linguistic information ob- tains a value BE[0, g and B#(O, -.g, then an approximation Definition 4 (Weighted average operator). Let x=I(1, a1),.... function is used to express the result in S. (rn, an)I be a set of linguistic 2-tuples and W=WI, .. wn be their associated weights. The 2-tuple weighted average x is: Definition 1 Herrera and Martinez, 2000. Let b be the result of an aggregation of the indexes of a set of labels assessed in a linguistic term set S, i.e. the result of a symbolic aggregation operation. (n,21)…、(m,=4(2=10nx)-=4( BE[O, g. Let i= round(B)and a=B-i be two values, such that, iE [O g] and a E[-5.5 )then a is called a Symbolic Translation The 2-tuple fuzzy linguistic approach is developed from the Definition 5(Linguistic weighted average operator). Let concept of symbolic translation by representing the linguistic x=[(r1,1),.,(Ta, n)) be a set of linguistic 2-tuples and nformation by means of 2-tuples (S, M),S E S and a E[-5.5) weights. The 2-tuple linguistic weighted average xr is S represents the linguistic label of the information, and a, is a numerical value expressing the value of the translation xW((r1, a1).(w1, a").((m, an), (wn, a-M))=4 B1·Bw from the original result B to the closest index label, i, in the lin guistic term set(s∈S) with B,=4"(r, a4)and Bw,=4"(w, a) This model defines a set of transformation functions between numeric values and 2-tuples. 2. 2.2. The multi-granular fuzzy linguistic modeling Definition 2(Herrera and Martinez, 2000). LetS=(So,., s be a In any fuzzy linguistic approach, an important parameter to determine is the" granularity of uncertainty", i.e. the cardinality linguistic term set and Be [0 g] a value representing the result of a of the linguistic term set S. According to the uncertainty degre ymbolic aggregation operation, then the 2-tuple that expresses that an expert qualifying a phenomenon has on it. the linguistic the equivalent information to B is obtained with the following term set chosen to provide his knowledge will have more or less A:0.g→s×-0.50.5) ent granularity of uncertainty are necessary(herrera Martinez, 4(0=(s.x,wit5 i=round(e). 2001: Herrera-Viedma et al, 2005 ). The use of different labels sets la=B-i aE[-5.5) to assess information is also necessary when an expert has to as- sess different ts,as for example it happens in information where round()is the usual round operation, St has the closest index uate the importance of the query terms el to"p"and"a"is the value of the symbolic translation and the relevance of the retrieved documents(Herrera-Viedma et al., 2003). In such situations, we need tools to manage multi- For all 4 there exists 4", defined as 4"(st,a)=i+a. On the granular linguistic information. In(Herrera Martinez, 2001)a other hand, it is obvious that the conversion of a linguistic term multi-granular 2-tuple FLm based on the concept of linguistic hier into a linguistic 2-tuple consists of adding a symbolic translation archy is proposed. alue of o:s∈S→(S,0 A Linguistic Hierarchy, LH, is a set of levels I(t, n(t)), i.e., The computational model is defined by presenting the following LH=U l(t, n(t), where each level t is a linguistic term set with a different granularity n(t) from the remaining of levels of the hier- archy. The levels are ordered according to their granularity, i.e.a (1)Negation operator: Neg((s, ax))=4(g-(4"(5,, ax) level t 1 provides a linguistic refinement of the previous level t. (2)Comparison of 2-tuples(Sk, a,)and(s1, a2): We can define a level from its predecessor level as: I(t,n(t)) If k <I then(Sk, 1) is smaller than(s,, a2) I(t +1, 2. n(t)-1) Table 1 shows the granularity needed in each Ifk=l then linguistic term set of the level t depending on the value n(t) defined (a)if a1=a2 then(Sk, a1)and(S a2)represent the same in the first level(3 and pectively A graphical example of a linguistic hierarchy is shown in Fig. 1 (b)if a1 a2 then(Sk, a1)is smaller than(S, az). (c)if a1>%2 then(Sk, %,)is bigger than(S, a2). (3)Aggregation operators. The aggregation of information con- Table 1 sists of obtaining a value that summarizes a set of values, Linguistic hierarchies. therefore, the result of the aggregation of a set of 2-tuples must be a 2-tuple. In the literature we can find many aggre- gation operators which allow us to combine the information l(t, n(t)) I(2.5) according to different criteria Using functions 4 and 4-l (,n(r)(classical and ordinal (Herrera & Herrera-Viedma, 1997; Zadeh, 1975)). To define it we have to establish the 2-tuple representation model and the 2-tuple computational model to represent and aggregate the linguistic information, respectively. Let S ¼ fs0; ... ; sg g be a linguistic term set with odd cardinality, where the mid term represents a indifference value and the rest of the terms is symmetrically related to it. We assume that the semantics of the labels is given by means of triangular membership functions and we consider that all terms are distributed on a scale on which a total order is defined, si 6 sj () i 6 j. In this fuzzy lin￾guistic context, if a symbolic method (Herrera & Herrera-Viedma, 1997; Herrera et al., 1996) aggregating linguistic information ob￾tains a value b 2 ½0; g, and b R f0; ... ; gg; then an approximation function is used to express the result in S. Definition 1 Herrera and Martínez, 2000. Let b be the result of an aggregation of the indexes of a set of labels assessed in a linguistic term set S, i.e., the result of a symbolic aggregation operation, b 2 ½0; g. Let i ¼ roundðbÞ and a ¼ b  i be two values, such that, i 2 ½0; g and a 2[.5,.5 )then a is called a Symbolic Translation. The 2-tuple fuzzy linguistic approach is developed from the concept of symbolic translation by representing the linguistic information by means of 2-tuples ðsi; aiÞ; si 2 S and ai 2[.5,.5): si represents the linguistic label of the information, and ai is a numerical value expressing the value of the translation from the original result b to the closest index label, i, in the lin￾guistic term set ðsi 2 SÞ. This model defines a set of transformation functions between numeric values and 2-tuples. Definition 2 (Herrera and Martínez, 2000). Let S ¼ fs0; ... ; sg g be a linguistic term set and b 2 ½0; g a value representing the result of a symbolic aggregation operation, then the 2-tuple that expresses the equivalent information to b is obtained with the following function: D : ½0; g ! S  ½0:5; 0:5Þ DðbÞ¼ðsi; aÞ; with si i ¼ roundðbÞ; a ¼ b  i a 2 ½:5; :5Þ; where roundðÞ is the usual round operation, si has the closest index label to ‘‘b” and ‘‘a” is the value of the symbolic translation. For all D there exists D1 , defined as D1 ðsi; aÞ ¼ i þ a. On the other hand, it is obvious that the conversion of a linguistic term into a linguistic 2-tuple consists of adding a symbolic translation value of 0 : si 2 S ) ðsi; 0Þ. The computational model is defined by presenting the following operators: (1) Negation operator: Negððsi; aÞÞ ¼ Dðg  ðD1 ðsi; aÞÞÞ. (2) Comparison of 2-tuples ðsk; a1Þ and ðsl; a2Þ: If k < l then ðsk; a1Þ is smaller than ðsl; a2Þ. If k ¼ l then (a) if a1 ¼ a2 then ðsk; a1Þ and ðsl; a2Þ represent the same information, (b) if a1 < a2 then ðsk; a1Þ is smaller than ðsl; a2Þ, (c) if a1 > a2 then ðsk; a1Þ is bigger than ðsl; a2Þ. (3) Aggregation operators. The aggregation of information con￾sists of obtaining a value that summarizes a set of values, therefore, the result of the aggregation of a set of 2-tuples must be a 2-tuple. In the literature we can find many aggre￾gation operators which allow us to combine the information according to different criteria. Using functions D and D1 that transform without loss of information numerical values into linguistic 2-tuples and viceversa, any of the existing aggregation operator can be easily extended for dealing with linguistic 2-tuples. Some examples are: Definition 3 (Arithmetic mean). Let x ¼ fðr1; a1Þ; ... ;ðrn; anÞg be a set of linguistic 2-tuples, the 2-tuple arithmetic mean xe is computed as, xe ½ðr1; a1Þ; ... ;ðrn; anÞ ¼ D Xn i¼1 1 n D1 ðri; aiÞ ! ¼ D 1 n Xn i¼1 bi !: Definition 4 (Weighted average operator). Let x ¼ fðr1; a1Þ; ... ; ðrn; anÞg be a set of linguistic 2-tuples and W ¼ fw1; ... ; wng be their associated weights. The 2-tuple weighted average xw is: xw½ðr1; a1Þ; ... ;ðrn; anÞ ¼ D Pn i¼1D1 ðri P ; aiÞ  wi n i¼1wi ! ¼ D Pn i¼1bi P  wi n i¼1wi : Definition 5 (Linguistic weighted average operator). Let x ¼ fðr1; a1Þ; ... ;ðrn; anÞg be a set of linguistic 2-tuples and W ¼ fðw1; aw 1 Þ; ... ;ðwn; aw n Þg be their linguistic 2-tuple associated weights. The 2-tuple linguistic weighted average xw l is: xw l ½ððr1; a1Þ;ðw1; aw 1 ÞÞ    ððrn; anÞ;ðwn; aw n ÞÞ ¼ D Pn i¼1bi P  bWi n i¼1bWi !; with bi ¼ D1 ðri; aiÞ and bWi ¼ D1ðwi; aw i Þ. 2.2.2. The multi-granular fuzzy linguistic modeling In any fuzzy linguistic approach, an important parameter to determine is the ‘‘granularity of uncertainty”, i.e., the cardinality of the linguistic term set S. According to the uncertainty degree that an expert qualifying a phenomenon has on it, the linguistic term set chosen to provide his knowledge will have more or less terms. When different experts have different uncertainty degrees on the phenomenon, then several linguistic term sets with a differ￾ent granularity of uncertainty are necessary (Herrera & Martínez, 2001; Herrera-Viedma et al., 2005). The use of different labels sets to assess information is also necessary when an expert has to as￾sess different concepts, as for example it happens in information retrieval problems, to evaluate the importance of the query terms and the relevance of the retrieved documents (Herrera-Viedma et al., 2003). In such situations, we need tools to manage multi￾granular linguistic information. In (Herrera & Martínez, 2001) a multi-granular 2-tuple FLM based on the concept of linguistic hier￾archy is proposed. A Linguistic Hierarchy, LH, is a set of levels l(t,n(t)), i.e., LH ¼ S tlðt; nðtÞÞ, where each level t is a linguistic term set with a different granularity nðtÞ from the remaining of levels of the hier￾archy. The levels are ordered according to their granularity, i.e., a level t þ 1 provides a linguistic refinement of the previous level t. We can define a level from its predecessor level as: lðt; nðtÞÞ ! lðt þ 1; 2  nðtÞ  1Þ. Table 1 shows the granularity needed in each linguistic term set of the level t depending on the value n(t) defined in the first level (3 and 7, respectively). A graphical example of a linguistic hierarchy is shown in Fig. 1. Table 1 Linguistic hierarchies. Level 1 Level 2 Level 3 lðt; nðtÞÞ lð1; 3Þ lð2; 5Þ lð3; 9Þ lðt; nðtÞÞ lð1; 7Þ lð2; 13Þ 12522 C. Porcel et al. / Expert Systems with Applications 36 (2009) 12520–12528