A hybrid fuzzy-based Personalized Recommender System for Telecom Products/Services
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3 fuzzy Techniques PreliminaRIesFor the description of the proposed approach, based on Zadeh [35], we first introduce some basic notions of fuzzy sets, fuzzy numbers, positive and negative fuzzy numbers, linguistic variables etc., and give related theorems [36]. These notions are used in a linguistic term similarity calculation in the proposed recommendation approach and the FTCP-RS software. Definition 1 A fuzzy set  in a universe of discourse X is characterized by a membership function  which associates with each element x in X a real number in the interval [0, 1]. The function value is termed the grade of membership of x in . A fuzzy number  is a fuzzy set, which is defined in a set of all real numbers R.
Definition 2 The λ-cut of fuzzy number is defined
 (1)
where Definition 3 A triangular fuzzy number can be defined by a triplet  and the membership function  is defined as:
 . (2)
From Definition 3, we can deduce that  .
Definition 4 If is a fuzzy number and  for any  , then is called a positive fuzzy number. Let  be the set of all finite positive fuzzy numbers on R.
Definition 5 For any  and  ,
 (3)
 (4)
 (5)
Definition 6 Let and  be two fuzzy numbers. Then  if  and  for any .
Definition 7 A linguistic variable is a variable whose values are words or sentences in a natural or artificial language. A linguistic variable is characterized by a quintuple  in which  is the name of the variable;  is the term-set of , that is, the collection of its linguistic values;  is a universe of discourse;  is a syntactic rule which generates the terms in ; and  is a semantic rule which associates with each linguistic value  its meaning,  , where denotes a fuzzy subset of [37].
Definition 8 Let then the vertex method is defined to calculate the distance between them as
 (6)
Definition 9 Let  then fuzzy number is closer to fuzzy number as  approaches 0.
In this study, a set of five linguistic terms {Strongly Interested (SI), More Interested (MI), Interested (I), Less Interested (LI), Not Interested (NI)} are used to describe the user ratings. The related fuzzy numbers to these linguistic terms are shown in Table 2. Their membership functions are illustrated in Figure 1. Table 2. Linguistic terms and related fuzzy numbers
Figure 1. Fuzzy sets and membership functions for Table 2 These definitions will be used in the following sections.
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is a nonempty bounded closed interval contained in X and it can be denoted by 
, 
and 
are the lower and upper bounds of the closed interval, respectively.