Revisiting KM algorithms: A Linear Programming approach

Computing the centroid and performing Type Reduction (TR) for type-2 fuzzy sets and systems are operations that must be taken into consideration. Karnik-Mendel Algorithms (KMAs) have been usually employed to perform these operations. In KMAs, these operations are defined as nonlinear optimization problems which are solved iteratively by finding the optimal Switching Points (SPs). In this study, we will transform these operations into Linear Fractional Programming (LFP) problems and solve them with the aids of the well-developed Linear Programming (LP) theory. It will be shown that there exists a direct relationship between the SPs of the KMAs and the solution vectors of the defined LFP problems. Thus, the meaning of the SPs will be revealed in the framework of LFP theory and the KMA will be connected a LFP method. We will then present two novel LP based TR methods which only use and employ basic built-in LP functions. Thus, these LP based TR methods will be very helpful in employing type-2 fuzzy sets and systems in different programming languages. Moreover, by taking account the connection of LFP to KMAs, a computationally efficient LP based TR method will be proposed. It will be proven that this LP based TR method can be seen as a kind of variation of the KMA (or vice versa). Simulation results have been presented to show the superiority of the LP based TR method in comparison to the KMA and Enhanced KMA.