Construction of slope-consistent trapezoidal interval type-2 fuzzy sets for simplifying the perceptual reasoning method

Computing with words (CWW) proposed by Zadeh is an useful paradigm to mimic the human decision-making ability in a wide variety of physical and mental tasks. To realize CWW, Mendel proposed a specific architecture called perceptual computer, in which interval type-2 (IT2) fuzzy sets (FSs) and perceptual reasoning (PR) method are adopted. The PR method has been proved to have good properties (e.g. it can output intuitive IT2 FSs) and has found several applications in decision making. In this study, we focus on simplifying this method by avoiding its a-cuts based inference process. We first present a novel property for the inference of the PR method. We observe from the property that, if the IT2 FSs in the consequents of the IF-THEN rules are trapezoidal and have consistent slopes, then the output IT2 FS will be strictly trapezoidal and can be determined easily. In this case, the computation of the PR method can be simplified. To achieve such simplification, the trapezoidal IT2 FSs without consistent slopes should be approximated by the slope-consistent trapezoidal IT2 FSs. This issue is also studied in this paper by solving the constrained linear-quadratic optimization problem. At last, examples are given. The simplified PR method will be useful when the CWW models are utilized in the modeling and/or control problems of complex systems or multivariable dynamic systems.