## 1 Introduction  
Nowadays, Type-1 and Type-2 fuzzy logic have demonstrated many advantages with respect to classical modeling methods, for example, their robustness, and design without a plant mathematical model. These advantages allow a relatively easy implementation in specific applications, for example, [1; 2] provide a review of control applications, [3] is an overview of pattern recognition and classification, [4; 5; 6] details hardware implementations of fuzzy systems, image processing [7], and many other applications as mentioned in [8]. However, the implementation of these algorithms usually requires a higher computational cost with respect to their classical counter parts.  
The computational cost of Interval Type-2 fuzzy logic was increased with respect to Type-1 fuzzy logic, of course achieving better results, but at the expense of having a higher computational cost. The implementation of Interval Type-2 fuzzy inference systems (FISs) requires around double the computational cost than Type-1 fuzzy inference systems, but could be even higher because the complexity of the centroid calculation increases in IT2 FISs, and the process to obtain the centroid of IT2 FIS is called type-reduction [9].  
The original type-reduction method was proposed by [PERSON] in [PERSON] (2010) and was called the Karnik-Mendel Algorithm (KM), and based on this method many variations have emerged, like the Enhanced Karnik-Mendel (EKM) [PERSON] and [PERSON] (2010), Continuous Karnik-Mendel (CKM), and Continuous Enhanced Karnik-Mendel (CEKM). These variations of the KM type reductions were proposed in order to reduce the type-reduction computational cost; and this is because in an IT2 FIS the process that requires the highest computational cost is in fact the type reduction.  
The goal of decreasing the type-reduction computational cost have been considered by several researchers, for example, in [PERSON] and [PERSON] (2010) the need for sorting is eliminated, in [PERSON] and [PERSON] (2010) a linear approximation of KM type reduction was realized, in [PERSON] and [PERSON] (2010) a polynomial regression for type reduction is realized, [PERSON] and [PERSON] (2010) introduces the Nie-Tan type reduction as an alternative to reduce the computational cost of this process by eliminating the type-reduction process, and [PERSON] and [PERSON] (2010) provides an study of several type-reduction alternatives focused on computational cost reduction.  
The contribution of the present work is a proposed new CEKM type-reduction approximation and also providing a methodology for its implementation. This method is based on CEKM as root-finding problem, which was proposed by [PERSON] and [PERSON] in [PERSON] and [PERSON] (2010). This method for type reduction is based on two proposed equations that allow approximation of the output points through numerical methods, like [PERSON] (2009). The objective is to provide a solution of type-reduction/defuzzification problem that eliminates the iterations and sampling requirements. In addition, we compare this approximation method with respect to the classical type-reductions approaches and propose in which context is recommendable to use our approach.  
The organization of the present work is as follows: Section 2 introduces the concepts of Interval Type-2 fuzzy inference systems, then Section 3 is focused on the type-reduction process, Section 4 presents the fundamentals of the proposed approximation of type reduction, Section 5 introduces the proposed methodology to realize the type-reduction approximation, Sections 6 and 7 outline two applications of our proposed methodology and their corresponding results, and finally Section 8 contains the conclusion and future work.