C Porcel et aL /Expert Systems with Applications 36(2009)5173-518 5175 matically update the user profile(Hanani et al, 2001; Popescul et BE[0. g. Let i=round() and a=B-i be two values, such that, aL., 2001; Reisnick& Varian, 1997) iE[0, g and aE[-5,5) then a is called a Symbolic Translation. Another important aspect that we must have in mind when we design a recommender system is the method to gather user infor The 2-tuple fuzzy linguistic approach is developed from the mation. In order to discriminate between relevant and irrelevant concept of symbolic translation by representing the linguistic information for a user, we must have some information about this information by means of 2-tuples (S, x),s ndx∈|-5,5 user,i.e, we must know the user preferences. Information about user preferences can be obtained in two different ways(Hanani S represents the linguistic label of the information, and et al., 2001; Quiroga Mostafa, 2002), implicit and explicit mode, z is a numerical value expressing the value of the translation although these ways not be mutually exclusive. from the original result B to the closest index label, i, in the lin- The implicit approach is implemented by inference from ustic term set(s∈S kind of observation. The observation is applied to user behavior or to detecting a user's environment(such as bookmarks or visited This model defines a set of transformation functions between URL). The user preferences are updated by detecting changes while numeric values and 2-tuples. bserving the user. On the other hand, the explicit approach, inter- Definition 2.(Herrera 8 Martinez, 2000). Let S=(So,., sgl be a acts with the users by acquiring feedback on information that is fil- linguistic term set and B E[O g] a value representing the result of a tered, that is, the user expresses some specifications of what they symbolic aggregation operation, then the 2-tuple that expresses desire. This last approach is very used(Hanani et al., 2001: the equivalent information to B is obtained with the following Popescul et al, 2001: Reisnick Varian, 1997). function: 2. 2. Fuzzy linguistic modeling A:10.g]→S×[0505) There are situations in which the information cannot be as- 4(B)=(Si, az),with i= round(B x=B-ia∈|-.5.5) sessed precisely in a quantitative form but may be in a qualitative one. For example, when attempting to quality phenomena related where round() is the usual round operation, s, has the closest index to human perception, we are often led to use words in natural lan- label to"Band"o" is the value of the symbolic translation. guage instead of numerical values. In other cases, precise quantita- tive information cannot be stated because either it is unavailable or For all 4 there exists A-l, defined as 4-(si, a)=i+a. On the the cost for its computation is too high and an approximate value other hand, it is obvious that the conversion of a linguistic term can be applicable The use of Fuzzy Sets Theory has given very good into a linguistic 2-tuple consists of adding a symbolic translation results for modeling qualitative information(Zadeh, 1975)and it value of 0: S:ES=(s4,0) has proven to be useful in many problems, e.g., in decision making The computational model is defined by presenting the following (Herrera, Herrera-Viedma, Verdegay, 1996: Herrera et al, 1996: operators: Herrera, Herrera-Viedma, Verdegay, 1998: Xu, 2006), quality Herrera-Viedma Peis, 2003: Herrera-Viedma, Peis, Morales del 2. Comparison of 2-tuples(Sk, a,)and(S, 42): s a)) evaluation(Herrera-Vviedma, Pasi, Lopez-Herrera, Porcel, 2006: 1. Negation operator: Neg((si, a))=4(g-(4"( astillo, Alonso, Anaya, 2007), information retrieval(Herrera- Viedma, 2001; Herrera-Viedma, 2001: Herrera-Viedma lopez Ifk=I then Herrera, 2007: Herrera-Viedma, Lopez-Herrera, Luque, porcel )if 1=2 then (Sk, a1)and (S a2) represent the same 2007: Herrera-Viedma, Lopez-Herrera, Porcel, 2005), political information analysis(Arf, 2005), etc. It is a tool based on the concept of linguis (b)if a1 a2 then(Sk, a,)is smaller than(S,%) tic variable proposed by Zadeh(1975). Next we analyze the two ap- (c)if a,>a2 then(Sk, a1) is bigger than(S 2). proaches of FLM that we use in our system. 3. Aggregation operators. The aggregation of information consists of obtaining a value that summarizes a set of values, therefore, 2. 1. The 2-tuple fuzzy linguistic approach the result of the aggregation of a set of 2-tuples must be a 2-tuple. The 2-tuple FLM(Herrera Martinez, 2000) is a continuous In the literature we can find many aggregation operators which model of representation of information that allows to reduce the allow us to combine the information according to different crite- loss of information typical of other fuzzy linguistic approaches ria. Using functions 4 and 4 that transform without loss of (classical and ordinal Herrera Herrera-Viedma, 1997: Zadeh, nformation numerical values into linguistic 2-tuples and vice 1975). To define it we have to establish the 2-tuple representation versa, any of the existing aggregation operator can be easily model and the 2-tuple computational model to represent and extended for dealing with linguistic 2-tuples. Some examples are aggregate the linguistic information, respectively. Let S=(so,.,Ssl be a linguistic term set with odd cardinality, Definition 3. Arithmetic mean: Let x=((r1, a1)..... (rn, n)be a set where the mid term represents a indifference value and the rest of of linguistic 2-tuples, the 2-tuple arithmetic mean xe is computed the terms are symmetric relate to it. we assume that the semanti of labels is given by means of triangular membership functions and consider all terms distributed on a scale on which a total order is defined, S <S<i<j. In this fuzzy linguistic context, if a sym- bolic method(Herrera Herrera-Viedma, 1997: Herrera et al 1996)aggregating linguistic information obtains a value BE[0,g]. Definition 4. Weighted average operator: Let x=((r1,a1),., g), then an approximation function is used to ex-(a, a.) be a set of linguistic 2-tuples and w= wi,..., wn) be thei press the result in S. associated weights. The 2-tuple weighted average xis Definition 1.(Herrera E Martinez, 2000 ) Let B be the result of an aggregation of the indexes of a set of labels assessed in a linguistic Tn,an)=4 erm set S, i.e, the result of a symbolic aggregation operation,matically update the user profile (Hanani et al., 2001; Popescul et al., 2001; Reisnick & Varian, 1997). Another important aspect that we must have in mind when we design a recommender system is the method to gather user infor￾mation. In order to discriminate between relevant and irrelevant information for a user, we must have some information about this user, i.e., we must know the user preferences. Information about user preferences can be obtained in two different ways (Hanani et al., 2001; Quiroga & Mostafa, 2002), implicit and explicit mode, although these ways not be mutually exclusive. The implicit approach is implemented by inference from some kind of observation. The observation is applied to user behavior or to detecting a user’s environment (such as bookmarks or visited URL). The user preferences are updated by detecting changes while observing the user. On the other hand, the explicit approach, inter￾acts with the users by acquiring feedback on information that is fil￾tered, that is, the user expresses some specifications of what they desire. This last approach is very used (Hanani et al., 2001; Popescul et al., 2001; Reisnick & Varian, 1997). 2.2. Fuzzy linguistic modeling There are situations in which the information cannot be as￾sessed precisely in a quantitative form but may be in a qualitative one. For example, when attempting to qualify phenomena related to human perception, we are often led to use words in natural lan￾guage instead of numerical values. In other cases, precise quantita￾tive information cannot be stated because either it is unavailable or the cost for its computation is too high and an approximate value can be applicable. The use of Fuzzy Sets Theory has given very good results for modeling qualitative information (Zadeh, 1975) and it has proven to be useful in many problems, e.g., in decision making (Herrera, Herrera-Viedma, & Verdegay, 1996; Herrera et al., 1996; Herrera, Herrera-Viedma, & Verdegay, 1998; Xu, 2006), quality evaluation (Herrera-Viedma, Pasi, López-Herrera, & Porcel, 2006; Herrera-Viedma & Peis, 2003; Herrera-Viedma, Peis, Morales del Castillo, Alonso, & Anaya, 2007), information retrieval (Herrera￾Viedma, 2001; Herrera-Viedma, 2001; Herrera-Viedma & López￾Herrera, 2007; Herrera-Viedma, López-Herrera, Luque, & Porcel, 2007; Herrera-Viedma, López-Herrera, & Porcel, 2005), political analysis (Arfi, 2005), etc. It is a tool based on the concept of linguis￾tic variable proposed by Zadeh (1975). Next we analyze the two ap￾proaches of FLM that we use in our system. 2.2.1. The 2-tuple fuzzy linguistic approach The 2-tuple FLM (Herrera & Martı´ nez, 2000) is a continuous model of representation of information that allows to reduce the loss of information typical of other fuzzy linguistic approaches (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 are symmetric relate to it. We assume that the semantics of labels is given by means of triangular membership functions and consider all terms distributed on a scale on which a total order is defined, si 6 sj () i 6 j. In this fuzzy linguistic context, if a sym￾bolic method (Herrera & Herrera-Viedma, 1997; Herrera et al., 1996) aggregating linguistic information obtains a value b 2 ½0; g, and b R f0; ... ; gg; then an approximation function is used to ex￾press the result in S. Definition 1. (Herrera & Martı´nez, 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 ½:5; :5Þ 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 ½:5; :5Þ: 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 (si 2 S). This model defines a set of transformation functions between numeric values and 2-tuples. Definition 2. (Herrera & Martı´nez, 2000). Let S ¼ fs0; ... ; sgg 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 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 aggregation operators which allow us to combine the information according to different crite￾ria. 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 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 i¼1 wi ! ¼ D Pn i¼1bi P  wi n i¼1 wi : C. Porcel et al. / Expert Systems with Applications 36 (2009) 5173–5183 5175