A commentary on some of the intrinsic differences between grey systems and fuzzy systems

The aim of this paper is to distinguish between some of the more intrinsic differences that exist between grey system theory (GST) and fuzzy system theory (FST). There are several aspects of both paradigms that are closely related, it is precisely these close relations that will often result in a misunderstanding or misinterpretation. The subtly of the differences in some cases are difficult to perceive, hence why a definitive explanation is needed. This paper discusses the divergences and similarities between the interval-valued fuzzy set and grey set, interval and grey number; for both the standard and the generalised interpretation. A preference based analysis example is also put forward to demonstrate the alternative in perspectives that each approach adopts. It is believed that a better understanding of the differences will ultimately allow for a greater understanding of the ideology and mantras that the concepts themselves are built upon. By proxy, describing the divergences will also put forward the similarities. We believe that by providing an overview of the facets that each approach employs where confusion may arise, a thorough and more detailed explanation is the result. This paper places particular emphasis on grey system theory, describing the more intrinsic differences that sets it apart from the more established paradigm of fuzzy system theory

A comparative study of fuzzy logic controllers for autonomous robots.

This paper presents the results from an experimental comparison of a number of fuzzy logic controllers performing mobile robot navigation. An experiment is described which requires the robot to complete a complex, measurable navigational task. The world's first generalised type-2 fuzzy logic controller is compared to a type-1 and a type-2 interval controller. Visual and statistical analyses show that the generalised type-2 fuzzy controller gives a better performance in consistency and smoothness. The impact of this paper comes primarily from the rigorous use of statistical analysis in mobile robot navigation

Quantification of Perception Clusters Using R-Fuzzy Sets and Grey Analysis

This paper investigates the use of the R-fuzzy significance measure hybrid approach introduced by the authors in a previous work; used in conjunction with grey analysis to allow for further inferencing, providing a higher dimension of accuracy and understanding. As a single observation can have a multitude of different perspectives, choosing a single fuzzy value as a representative becomes problematic. The fundamental concept of an R-fuzzy set is that it allows for the collective perception of a populous, and also individualised perspectives to be encapsulated within its membership set. The introduction of the significance measure allowed for the quantification of any membership value contained within any generated R-fuzzy set. Such is the pairing of the significance measure and the R-fuzzy concept, it replicates in part, the higher order of complex uncertainty which can be garnered using a type-2 fuzzy approach, with the computational ease and objectiveness of a typical type-1 fuzzy set. This paper utilises the use of grey analysis, in particular, the use of the absolute degree of grey incidence for the inspection of the sequence generated when using the significance measure, when quantifying the degree of significance fore each contained fuzzy membership value. Using the absolute degree of grey incidence provides a means to measure the metric spaces between sequences. As the worked example will show, if the data contains perceptions from clusters of cohorts, these clusters can be compared and contrasted to allow for a more detailed understanding of the abstract concepts being modelled

An investigation into alternative methods for the defuzzification of an interval type-2 fuzzy set.

The number of applications of interval type-2 fuzzy logic to real world problems is growing. To date such systems have used type-reduction or an approximation of typereduction to arrive at final crisp output from the system. This paper describes the novel direct defuzzifier for interval type-2 fuzzy sets. A number of fuzzy systems are compared with direct defuzzifier. We compare each defuzzifiers output surface in a simple rule-based system. The direct defuzzifier compares favourably with the type-reduction method under the minimum t-norm. Also, we found that the non-stationary approach provided interesting results

Inventory optimisation with an interval type-2 fuzzy model.

The planning of resources within a supply chain can prove to be a deciding factor in the success or failure of an operation. This research continues the authors' previous work using an extended Interval Type-2 Fuzzy Logic supply chain model, with an Evolutionary Algorithm to search for good resource plans. A set of enhanced experiments is conducted to validate our novel approach with optimal configurations, and determine an appropriate Evolutionary Algorithm set up for the given problem

A Significance Measure for R-Fuzzy Sets

This paper presents a newly created significance measure based on a variation of Bayes' theorem, one which quantifies the significance of any value contained within an R-fuzzy set. An R-fuzzy set is a relatively new concept and an extension to fuzzy sets. By utilising the lower and upper approximations from rough set theory, an R-fuzzy approach allows for uncertain fuzzy membership values to be encapsulated. The membership values associated with the lower approximation are regarded as absolute truths, whereas the values associated with the upper approximation maybe be the result of a single voter, or the vast majority, but definitely not all. By making use of the significance measure one can inspect each and every encapsulated membership value. The significance value itself is a coefficient, this value will indicate how strongly it was agreed upon by the populace for a specific R-fuzzy descriptor. There has been no recent effort made in order to make sense of the significance of any of the values contained within an R-fuzzy set, hence the motivation for this paper. Also presented is a worked example, demonstrating the coupling together of an R-fuzzy approach and the significance measure

R-fuzzy Sets and Grey System Theory

This paper investigates the use of grey theory to enhance the concept of an R-fuzzy set, with regards to the precision of the encapsulating set of returned significance values. The use of lower and upper approximations from rough set theory, allow for an R-fuzzy approach to encapsulate uncertain fuzzy membership values; both collectively generic and individually specific. The authors have previously created a significance measure, which when combined with an R-fuzzy set provides one with a refined approach for expressing complex uncertainty. This pairing of an R-fuzzy set and the significance measure, replicates in part, the high detail of uncertainty representation from a type-2 fuzzy approach, with the relative ease and objectiveness of a type-1 fuzzy approach. As a result, this new research method allows for a practical means for domains where ideally a generalised type-2 fuzzy set is more favourable, but ultimately unfeasible due to the subjectiveness of type-2 fuzzy membership values. This paper focuses on providing a more effective means for the creation of the set which encapsulates the returned degrees of significance. Using grey techniques, rather than the arbitrary configuration of the original work, the result is a high precision set for encapsulation, with the minimal configuration of parameter values. A worked example is used to demonstrate the effectiveness of using grey theory in conjunction with R-fuzzy sets and the significance measure

Alpha-Level Aggregation: A Practical Approach to Type-1 OWA Operation for Aggregating Uncertain Information with Applications to Breast Cancer Treatments

Type-1 Ordered Weighted Averaging (OWA) operator provides us with a new technique for directly aggregating uncertain information with uncertain weights via OWA mechanism in soft decision making and data mining, in which uncertain objects are modeled by fuzzy sets. The Direct Approach to performing type-1 OWA operation involves high computational overhead. In this paper, we define a type-1 OWA operator based on the \alpha-cuts of fuzzy sets. Then, we prove a Representation Theorem of type-1 OWA operators, by which type-1 OWA operators can be decomposed into a series of \alpha-level type-1 OWA operators. Furthermore, we suggest a fast approach, called Alpha-Level Approach, to implementing the type-1 OWA operator. A practical application of type-1 OWA operators to breast cancer treatments is addressed. Experimental results and theoretical analyses show that: 1) the Alpha-Level Approach with linear order complexity can achieve much higher computing efficiency in performing type-1 OWA operation than the existing Direct Approach, 2) the type-1 OWA operators exhibit different aggregation behaviors from the existing fuzzy weighted averaging (FWA) operators, and 3) the type-1 OWA operators demonstrate the ability to efficiently aggregate uncertain information with uncertain weights in solving real-world soft decision-making problems

Type-2 fuzzy elliptic membership functions for modeling uncertainty

Whereas type-1 and type-2 membership functions (MFs) are the core of any fuzzy logic system, there are no performance criteria available to evaluate the goodness or correctness of the fuzzy MFs. In this paper, we make extensive analysis in terms of the capability of type-2 elliptic fuzzy MFs in modeling uncertainty. Having decoupled parameters for its support and width, elliptic MFs are unique amongst existing type-2 fuzzy MFs. In this investigation, the uncertainty distribution along the elliptic MF support is studied, and a detailed analysis is given to compare and contrast its performance with existing type-2 fuzzy MFs. Furthermore, fuzzy arithmetic operations are also investigated, and our finding is that the elliptic MF has similar features to the Gaussian and triangular MFs in addition and multiplication operations. Moreover, we have tested the prediction capability of elliptic MFs using interval type-2 fuzzy logic systems on oil price prediction problem for a data set from 2nd Jan 1985 till 25th April 2016. Throughout the simulation studies, an extreme learning machine is used to train the interval type-2 fuzzy logic system. The prediction results show that, in addition to their various advantages mentioned above, elliptic MFs have comparable prediction results when compared to Gaussian and triangular MFs. Finally, in order to test the performance of fuzzy logic controller with elliptic interval type-2 MFs, extensive real-time experiments are conducted for the 3D trajectory tracking problem of a quadrotor. We believe that the results of this study will open the doors to elliptic MFs’ wider use of real-world identification and control applications as the proposed MF is easy to interpret in addition to its unique features