Monotone Interval Fuzzy Inference Systems

—In this paper, we introduce the notion of a monotone fuzzy partition, which is useful for constructing a monotone zeroorder Takagi–Sugeno–Kang Fuzzy Inference System (ZOTSKFIS). It is known that a monotone ZOTSK-FIS model can always be produced when a consistent, complete, and monotone fuzzy rule base is used. However, such an ideal situation is not always available in practice, because a fuzzy rule base is susceptible to uncertainties, e.g., inconsistency, incompleteness, and nonmonotonicity. As a result, we devise an interval method to model these uncertainties by considering the minimum interval of acceptability of a fuzzy rule, resulting in a set of monotone interval-valued fuzzy rules. This further leads to the formulation of a Monotone Interval Fuzzy Inference System (MIFIS) with a minimized uncertainty measure. The proposed MIFIS model is analyzed mathematically and evaluated empirically for the Failure Mode and Effect Analysis (FMEA) application. The results indicate that MIFIS outperforms ZOTSK-FIS, and allows effective decision making using uncertain fuzzy rules solicited from human experts in tackling real-world FMEA problems

Parametric Conditions for a Monotone TSK Fuzzy Inference System to be an n-Ary Aggregation Function

Despite the popularity and practical importance of the fuzzy inference system (FIS), the use of an FIS model as an n-ary aggregation function, which is characterized by both the monotonicity and boundary properties, is yet to be established. This is because research on ensuring that FIS models satisfy the monotonicity property, i.e., monotone FIS, is relatively new, not to mention the additional requirement of satisfying the boundary property. The aim of this article, therefore, is to establish the parametric conditions for the Takagi–Sugeno–Kang (TSK) FIS model to operate as an n-ary aggregation function (hereafter denoted as n-TSK-FIS) via the specifications of fuzzy membership functions and fuzzy rules. An absorption property with fuzzy rules interpretation is outlined, and the use of n-TSK-FIS as a uninorm is explained. Exploiting the established parametric conditions, a framework for which an n-TSK-FIS model can be constructed from data samples is formulated and analyzed, along with a number of remarks. Synthetic data sets and a benchmark example on education assessment are presented and discussed. To be best of the authors’ knowledge, this article serves as the first use of the TSK-FIS model as an n-ary aggregation function

Parametric Conditions for a Monotone TSK Fuzzy Inference System to be an n-Ary Aggregation Function

Despite the popularity and practical importance of the Fuzzy Inference System (FIS), the use of an FIS model as an n -ary aggregation function, which is characterized by both the monotonicity and boundary properties, is yet to be established. This is because research on ensuring that FIS models satisfy the monotonicity property, i.e., monotone FIS, is relatively new, not to mention the additional requirement of satisfying the boundary property. The aim of this paper, therefore, is to establish the parametric conditions for the Takagi-Sugeno-Kang (TSK) FIS model to operate as an n -ary aggregation function (hereafter denoted as n -TSK-FIS) via the specifications of fuzzy membership functions (FMFs) and fuzzy rules. An absorption property with fuzzy rules interpretation is outlined, and the use of n -TSK-FIS as a uni-norm is explained. Exploiting the established parametric conditions, a framework for which an n -TSK-FIS model can be constructed from data samples is formulated and analyzed, along with a number of remarks. Synthetic data sets and a benchmark example on education assessment are presented and discussed. To be best of the authors' knowledge, this study serves as the first use of the TSK-FIS model as an n -ary aggregation function

An analytical interval fuzzy inference system for risk evaluation and prioritization in failure mode and effect analysis

The fuzzy inference system (FIS) is useful for developing an improved Risk Priority Number (RPN) model for risk evaluation in failure mode and effect analysis (FMEA). A general FIS_RPN model considers three risk factors, i.e., severity, occurrence, and detection, as the inputs and produces an FIS_RPN score as the output. At present, there are two issues pertaining to practical implementation of classical FIS_RPN models as follows: 1) the fulfillment of the monotonicity property between the FIS_RPN score (output) and the risk factors (inputs); and 2) difficulty in obtaining a complete and monotone fuzzy rule base. The aim of this paper is to propose a new analytical interval FIS_RPN model to solve the aforementioned issues. Specifically, the incomplete and potentially nonmonotone fuzzy rules provided by FMEA users are transformed into a set of interval-valued fuzzy rules in order to produce an interval FIS_RPN model. The interval FIS_RPN model aggregates a set of risk ratings and produces a risk interval, which is useful for risk evaluation and prioritization. Properties of the proposed interval FIS_RPN model are analyzed mathematically. An FMEA procedure that incorporates the proposed interval FIS_RPN model is devised. A case study with real information from a semiconductor company is conducted to evaluate the usefulness of the proposed model. The experimental results indicate that the interval FIS_RPN model is able to appropriately rank the failure modes, even when the fuzzy rules provided by FMEA users are incomplete and nonmonotone

A New Interval-based Method for Handling Non-Monotonic Information

The focus of this paper is on handling non-monotone information in the modelling process of a single-input target monotone system. On one hand, the monotonicity property is a piece of useful prior (or additional) information which can be exploited for modelling of a monotone target system. On the other hand, it is difficult to model a monotone system if the available information is not monotonically-ordered. In this paper, an interval-based method for analysing non-monotonically ordered information is proposed. The applicability of the proposed method to handling a non-monotone function, a non-monotone data set, and an incomplete and/or non-monotone fuzzy rule base is presented. The upper and lower bounds of the interval are firstly defined. The region governed by the interval is explained as a overage measure. The coverage size represents uncertainty pertaining to the available information. The proposed approach constitutes a new method to transform non-monotonic information to interval-valued monotone system. The proposed interval-based method to handle an incomplete and/or non-monotone fuzzy rule base constitutes a new fuzzy reasoning approach

Monotone Fuzzy Rule Interpolation for Practical Modeling of the Zero-Order TSK Fuzzy Inference System

Formulating a generalized monotone fuzzy rule interpolation (MFRI) model is difficult. A complete and monotone fuzzy rule-base is essential for devising a monotone zero-order Takagi–Sugeno–Kang (TSK) fuzzy inference system (FIS) model. However, such a complete and monotone fuzzy rule-base is not always available in practice. In this article, we develop an MFRI modeling scheme for generating a monotone zero-order TSK FIS from a monotone and incomplete fuzzy rule-base. In our proposal, a monotone-ordered fuzzy rule-base that consists of the available fuzzy rules from a monotone and incomplete fuzzy rule-base and those derived from the MFRI reasoning is formed. We outline three important properties that the MFRI's deduced fuzzy rules should satisfy to ensure a monotone-ordered fuzzy rule-base. A Lagrangian function for the MFRI scheme, together with its Karush–Kuhn–Tucker optimality conditions, is formulated and analyzed. The key idea is to impose constraints that guide the MFRI inference outcomes. An iterative MFRI algorithm that adopts an augmented Lagrangian function is devised. The proposed MFRI algorithm aims to achieve an ε -optimality condition and to produce an ε -optimal solution, which is geared for practical applications. We apply the MFRI algorithm to a failure mode and effect analysis case study and a tanker ship heading regulation problem. The results indicate the effectiveness of MFRI for generating monotone TSK FRI models in tackling practical problems