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Despite the popularity and practical importance of Q1 5 the fuzzy inference system (FIS), the use of an FIS model as an 6 n-ary aggregation function, which is characterized by both the 7 monotonicity and boundary properties, is yet to be established. 8 This is because research on ensuring that FIS models satisfy the Q2 9 monotonicity property, i.e., monotone FIS, is relatively new, not 10 to mention the additional requirement of satisfying the boundary 11 property. The aim of this article, therefore, is to establish the 12 parametric conditions for the Takagi-Sugeno-Kang (TSK) FIS 13 model to operate as an n-ary aggregation function (hereafter de-14 noted as n-TSK-FIS) via the specifications of fuzzy membership

In this paper, the problem of maintaining the (global) monotonicity and local monotonicity properties between the input(s) and the output of an FIS model is addressed. This is known as the monotone fuzzy modeling problem. In our previous work, this problem has been tackled by developing some mathematical conditions for an FIS model to observe the monotonicity property. These mathematical conditions are used as a set of governing equations for undertaking FIS modeling problems, and have been extended to some advanced FIS modeling techniques. Here, we examine an alternative to the monotone fuzzy modeling problem by introducing a monotonicity index. The monotonicity index is employed as an approximate indicator to measure the fulfillment of an FIS model to the monotonicity property. It allows the FIS model to be constructed using an optimization method, or be tuned to achieve a better performance, without knowing the exact mathematical conditions of the FIS model to satisfy the monotonicity property. Besides, the monotonicity index can be extended to FIS modeling that involves the local monotonicity problem. We also analyze the relationship between the FIS model and its monotonicity property fulfillment, as well as derived mathematical conditions, using the Monte Carlo method.

Abstract-In this paper, ad hoc and system identification methods are used to generate fuzzy If-Then rules for a zeroorder Takagi-Sugeno-Kang (TSK) Fuzzy Inference System (FIS) using a set of multi-attribute monotone data. Convex and normal trapezoidal fuzzy sets, with a strong fuzzy partition strategy, is employed. Our analysis shows that even with multi-attribute monotone data, non-monotone fuzzy IfThen rules can be produced using an ad hoc method. The same observation can be made, empirically, using a system identification method, e.g., a derivative-based optimization method and the genetic algorithm. This finding is important for modeling a monotone FIS model, as the result shows that even with a "clean" data set pertaining to a monotone system, the generated fuzzy If-Then rules may need to be preprocessed, before being used for FIS modeling. As such, monotone fuzzy rule relabeling is useful. Besides that, a constrained non-linear programming method for FIS modelling is suggested, as a variant of the system identification method.

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