Revisiting Karnik–Mendel Algorithms in the framework of Linear Fractional Programming

Abstract

For Interval Type-2 (IT2) fuzzy sets and systems, calculating the centroid and performing type reduction are operations that must be taken into consideration. The Karnik–Mendel Algorithm (KMA) and its enhancements have been usually employed to perform these operations. In KMAs, these IT2 fuzzy operations are defined as nonlinear optimization problems which are solved iteratively by finding the optimal Switching Points (SPs). In this study, we will examine these optimization problems and will reveal the nature of KMAs in the framework of Linear Fractional Programming (LFP). In this context, we will firstly transform IT2 fuzzy operations into 0-1 LFP problems and then we will show that there exists a direct relationship between the SPs of the KMAs and the solution vectors of the defined LFP problems. Hence, the meaning of the SPs will be revealed in the framework of LFP theory. Thus, it will be concluded that the KMAs can be seen as LFP methods. Then, by taking account of the LFP representations, we will show that it is possible to perform the IT2 fuzzy operations via well-known LFP methods, namely the Charnes Cooper Transformation and the Dinkelbach's Algorithm (DA). It is believed that the presented LFP approaches will be very helpful in employing IT2 fuzzy sets and systems in different programming languages since they only use and employ built-in Linear Programming (LP) routines. However, since LP solvers are used, it will be concluded that the LFP based approaches are computationally intensive and thus might not be feasible for real-time applications. Therefore, we will present a computationally efficient implementation of presented the DA based method, namely the KMK Algorithm (KMKA). Then, through analytical investigations on the solution vectors and optimality conditions of the KMKA, it will be revealed that the KMA is algorithmic equivalent to the presented implementation of the DA based KMKA. Thus, it will be revealed that the commonly employed KMA and its enhancements can be seen as efficient implementations of the DA in the framework of LFP. Based on this connection, since the DA is an adaptation of the Newton method to a non-differentiable context, it will be also concluded that the discrete version of KMA can be also seen equivalent to a root-finding problem. We will present comparative numerical results to validate the analytical derivations and observations.