J. Serrano-Guerrero et al Information Sciences 181 (2011)1503-1516 alues by means of linguistic variables 52. Its application has been successful to different problems such as information retrieval [5,6, 22-24, 28, recommender systems [40, 41]. quality evaluation [26, 27, 36 decision making 8,9, 17.3 The 2-tuple FLM [20 is a continuous model of representation of information which reduces the loss of typical of other fuzzy linguistic approaches (classical and ordinal [ 16, 52)). To define it we have to establish representation model and the 2-tuple computational model to represent and aggregate the linguistic respectively. Let S=(So, J be a linguistic term set with odd cardinality, where the middle term an indifference value and the rest of the terms are symmetrically related to it. We assume that the semantics of the labels are given by means of tri- ngular membership functions and we consider that all terms are distributed on a scale in which a total order is defined, S,<S)+isj. In this fuzzy linguistic context, a symbolic method [ 19, 16] aggregating linguistic information obtains a value BE[O, g] and B#(O,.... g) then an approximation function is used to express the result in S. Definition 1. Let B be the result of an aggregation of the indexes of a set of labels assessed ustic term set S, i.e the result of a symbolic aggregation operation, BE[O, g. Let i= round(e)and a=B-i be two such that,i∈[0.g]and ZIE[-5.5) then a is called a Symbolic Translation 20 The 2-tuple fuzzy linguistic approach is developed from the concept of symbolic translation by representing the linguistic S: represents the linguistic label of the information, and xi is a numerical value expressing the value of the translation from the original result B to the closest index label, i, in the nguistic term set (SiE S). This model defines a set of transformation functions between numeric values and 2-tuples. Definition 2. Let S=(so. l,.,Sg) be a linguistic term set and BE[O, 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[20 △:[0.g]→S×-0.50.5), △(=(sxwt(5 i= round (e) x=B-ix∈-5,5 where round() is the usual round operation, s, has the closest index label to"Banda"is the value of the symboli For all A there exists A-(Si, ax)=i+ a. On the other hand, it is obvious that the conversion of a linguistic term into a lin- guistic 2-tuple consists of adding a symbolic translation value of 0: SES=(S, O). The computational model is defined by presenting the following operator 1. Negation operator: Neg(s, x)=A(g-A"(si, a)) 2. Comparison of 2-tuples(Sk, a,)and(S, a2) if k< I then(Sk, &1) is smaller than(Si, a2) if k=I then (a) if a1=%2 then(Sk,a,)and(sp,a2)represent th e same l (b) if a1< 2 then(Sk, o) is smaller than(S, a2) (c) if 1> a2 then(Sk, M) is bigger than(s,2) 3. Aggregation operators. The aggregation of information consists of obtaining a value that summarizes a set of values, herefore, the result of the aggregation of a set of 2-tuples must be a 2-tuple. In the literature we can find many aggre- tion operators which allow us to combine the information according to different criteria Using functions A and A hat transform without loss of information numerical values into linguistic 2-tuples and vice versa, any of the existing gregation operator can be easily extended for dealing with linguistic 2-tuples. Some examples are Definition 3(Arithmetic mean). Let x=((r1, a1),... (n, Mn)) be a set of linguistic 2-tuples, the 2-tuple arithmetic mean x is omputed as xn2x)…-(2(0)-4(S) Definition 4(Weighted average operator). Let x=((r1, 1)..... (rn, an) be a set of linguistic 2-tuples and w=(wi,., wn) be their associated weights. The 2-tuple weighted average xis:values by means of linguistic variables [52]. Its application has been successful to different problems such as information retrieval [5,6,22–24,28], recommender systems [40,41], quality evaluation [26,27,36], decision making [8,9,17,32,53], etc. The 2-tuple FLM [20] is a continuous model of representation of information which reduces the loss of information typical of other fuzzy linguistic approaches (classical and ordinal [16,52]). 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 = {s0,..., sg} be a linguistic term set with odd cardinality, where the middle term represents an indifference value and the rest of the terms are symmetrically related to it. We assume that the semantics of the labels are given by means of tri￾angular membership functions and we consider that all terms are distributed on a scale in which a total order is defined, si 6 sj () i 6 j. In this fuzzy linguistic context, a symbolic method [19,16] aggregating linguistic information obtains a value b 2 [0,g] and b R {0,...,g} then an approximation function is used to express the result in S. Definition 1. 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 [0,g]. Let i = round(b) and a = b i be two values, such that, i 2 [0,g] and ai 2 [.5, .5) then a is called a Symbolic Translation [20]. The 2-tuple fuzzy linguistic approach is developed from the concept of symbolic translation by representing the linguistic information by means of 2-tuples (si,ai), 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 linguistic term set (si 2 S). This model defines a set of transformation functions between numeric values and 2-tuples. Definition 2. Let S = {s0, s1,..., sg} be a linguistic term set and b 2 [0,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 [20]: D : ½0; g ! S  ½0:5; 0:5Þ; DðbÞ¼ðsi; aÞ; with si i ¼ roundðbÞ; a ¼ b i ai 2 ½:5; :5Þ; where round() 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 (si,a) = i + a. On the other hand, it is obvious that the conversion of a linguistic term into a lin￾guistic 2-tuple consists of adding a symbolic translation value of 0: si 2 S ) (si,0). The computational model is defined by presenting the following operators: 1. Negation operator: Neg(si,a) = D(g D1 (si,a)) 2. Comparison of 2-tuples (sk,a1) and (sl,a2): if k < l then (sk,a1) is smaller than (sl,a2) if k = l then (a) if a1 = a2 then (sk,a1) and (sl,a2) represent the same informations (b) if a1 6 a2 then (sk,a1) is smaller than (sl,a2) (c) if a1 P a2 then (sk,a1) is bigger than (sl,a2) 3. Aggregation operators. The aggregation of information consists 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 vice versa, 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 = {(r1,a1),..., (rn,an)} 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 = {(r1,a1),..., (rn,an)} be a set of linguistic 2-tuples and W = {w1,...,wn} be their associated weights. The 2-tuple weighted average xw is: 1506 J. Serrano-Guerrero et al. / Information Sciences 181 (2011) 1503–1516