A. Zenebe, A F. Norcio/ Fuzzy Sets and Systems 160(2009)76-94 Membership degree of movie to genres for User 7 Movie (lj) Gi (vector for jth movie) 0.683 1.000 0.000 0.000 4 0.438 0.000 1000 0.000 Based on the heuristics illustrated above, the possibility for item Ij to take different values of X varies and the membership function should meet the following four criteria: (1)assigning higher degree of membership to major values than minor values; (2) assigning O to values that are not associated with the item; (3)degrees of membership should be normalized to the range of [0, 1]; and(4)the same value of X at similar rank positions between different items should have varying degrees of membership values if the number of values of X associated with the items are different. We represent this type of heuristic with a Gaussian-like membership function, as shown in(1). Ax,(Ij)=rk/2v where N=Ilil is the number of values of X associated with Ij and rk(I<rk <ILiD is the rank position of value xk and a> I is a parameter used as a threshold to control the difference between consecutive values of X in/j. Moreover, is the only parameter that needs to be determined For example, using Eq(1)with a set to 1. 2(after various experimental trials), the movie 'King Kong(2005 0.211),(Thriller,.168)). Furthermore, for User 7, Table 1 shows the representation of some of the rated movies represented in terms of genres: for Lil= 5 and xj=[(Action, 1),(Adventure, 0.366), (Drama, 0. 272),(Fanta Ross et al. [23] stated that a membership function(MF)should match"the intuitively plausible semantic description of imprecise properties of the objects in X. "The reason we make the first important genre completely satisfied is movies are advertised exclusively as having a specific genre such as dram a and comedy; as well users have similar perception to movies. The soundness of the representation and inference method are further investigated in Section 5 using the movie dataset The reciprocal function defined as 1/k for k= l to N is also investigated. The problem with the reciprocal function is it does not consider the total number of different genres that exist in movies leading to the same degree of membership value for the same genre at same rank position with different number of genres. A uniform distribution membership of genres in a movie, assuming a movie with multiple genres has equal degree of genre presence for all occurring genres represented by l or 0, is the baseline crisp set representation. The heuristic that leads to the membership function in(1) is developed based on the analysis of the movie dataset literature on movies [10] and preliminary experimental trials conducted on a. This heuristic, for instance, assumes that two genres will not have equal degree of presence in a movie. That is, two or more genres cannot exist in"equal degree of presence"or"in equal amount"in a movie. For example, let us say a movie X has drama and comedy. The movie cannot have dramatic and comedy nature with exact equal amount, i.e. with exact degree of membership 0.5 each. This is also true in the movies database used in this research. This assumption is logical because a movie cannot have exactly the same"content"of two or more genres. If the amount of content is close then a needs to be tuned Moreover, in future research, studying various membership functions is needed in order to get an optimal fuzzy set based recommender system. The basis of our study is, provided that a membership function that satisfies the four criteria identified above. A is used, the recommendations by FTM will be better than crisp-set based approach. The example, one can use movie actresses/actors as the second attribute. The actors in a movie can be represented in a vco. a representation scheme can be extended to recommender systems based on a combination of multiple attributes. Fe A=a1, a2,..., ak) for K actors. The role or importance of an actor or actress ak in a movie mi can be represented by degree of membership associated with the fuzzy variable 'degree of role or importance. That is, A j=l(ak, Ha ( i) for k= I to k], where pa(lj) can be determined heuristically or using machine learningA. Zenebe, A.F. Norcio / Fuzzy Sets and Systems 160 (2009) 76–94 81 Table 1 Membership degree of movie to genres for User 7 Movie (Ij ) Gj (vector for j th movie) Rating xj1 xj2 xj3 … xj15 xj16 56 4 0.683 1.000 0.438 … 0.000 0.000 79 5 1.000 0.000 0.000 … 0.467 0.000 89 3 0.683 0.000 0.000 … 0.000 1.000 … … … … … …… … 254 2 0.438 0.000 1.000 … 0.000 0.000 Based on the heuristics illustrated above, the possibility for item Ij to take different values of X varies and the membership function should meet the following four criteria: (1) assigning higher degree of membership to major values than minor values; (2) assigning 0 to values that are not associated with the item; (3) degrees of membership should be normalized to the range of [0, 1]; and (4) the same value of X at similar rank positions between different items should have varying degrees of membership values if the number of values of X associated with the items are different. We represent this type of heuristic with a Gaussian-like membership function, as shown in (1). xk (Ij ) = rk/2 √∗|Lj|(rk−1) , (1) where N = |Lj | is the number of values of X associated with Ij and rk (1rk |Lj |) is the rank position of value xk, and  > 1 is a parameter used as a threshold to control the difference between consecutive values of X in Ij . Moreover,  is the only parameter that needs to be determined. For example, using Eq. (1) with  set to 1.2 (after various experimental trials), the movie ‘King Kong (2005)’ is represented in terms of genres: for |Lj | = 5 and xj = {(Action, 1), (Adventure, 0.366), (Drama, 0.272), (Fantasy, 0.211), (Thriller, 0.168)}. Furthermore, for User 7, Table 1 shows the representation of some of the rated movies. Ross et al. [23] stated that a membership function (MF) should match “the intuitively plausible semantic description of imprecise properties of the objects in X.” The reason we make the first important genre completely satisfied is movies are advertised exclusively as having a specific genre such as drama and comedy; as well users have similar perception to movies. The soundness of the representation and inference method are further investigated in Section 5 using the movie dataset. The reciprocal function defined as 1/k for k = 1 to N is also investigated. The problem with the reciprocal function is it does not consider the total number of different genres that exist in movies leading to the same degree of membership value for the same genre at same rank position with different number of genres. A uniform distribution membership of genres in a movie, assuming a movie with multiple genres has equal degree of genre presence for all occurring genres represented by 1 or 0, is the baseline crisp set representation. The heuristic that leads to the membership function in (1) is developed based on the analysis of the movie dataset, literature on movies [10] and preliminary experimental trials conducted on . This heuristic, for instance, assumes that two genres will not have equal degree of presence in a movie. That is, two or more genres cannot exist in “equal degree of presence” or “in equal amount” in a movie. For example, let us say a movie X has drama and comedy. The movie cannot have dramatic and comedy nature with exact equal ‘amount’, i.e. with exact degree of membership 0.5 each. This is also true in the movies database used in this research. This assumption is logical because a movie cannot have exactly the same “content” of two or more genres. If the amount of content is close then  needs to be tuned. Moreover, in future research, studying various membership functions is needed in order to get an optimal fuzzy set based recommender system. The basis of our study is, provided that a membership function that satisfies the four criteria identified above. A is used, the recommendations by FTM will be better than crisp-set based approach. The representation scheme can be extended to recommender systems based on a combination of multiple attributes. For example, one can use movie actresses/actors as the second attribute. The actors in a movie can be represented in a vector A = {a1, a2,...,ak} for K actors. The role or importance of an actor or actress ak in a movie mi can be represented by degree of membership associated with the fuzzy variable ‘degree of role or importance’. That is, Aj = {(ak, ak (Ij )), for k = 1 to K}, where ak (Ij ) can be determined heuristically or using machine learning.