Improving the efficiency of type-reduction algorithms continues to attract research interest. Recently, there has been some new type-reduction approaches claiming that they are more efficient than the well-known algorithms such as the enhanced Karnik–Mendel (EKM) and the enhanced iterative algorithm with stopping condition (EIASC). In a previous paper, we found that the computational efficiency of an algorithm is closely related to the platform, and how it is implemented. In computer science, the dependence on languages is usually avoided by focusing on the complexity of algorithms (using big O notation). In this article, the main contribution is the proposal of two novel type-reduction algorithms. Also, for the first time, a comprehensive study on both existing and new type-reduction approaches is made based on both algorithm complexity and practical computational time under a variety of programming languages. Based on the results, suggestions are given for the preferred algorithms in different scenarios depending on implementation platform and application context.

In many contexts, type-2 fuzzy sets (T2 FS) are obtained from a type-1 fuzzy set to which we wish to add uncertainty. However, in the current type-2 representation, there is no restriction on the shape of the footprint of uncertainty and the embedded sets (ESs) that can be considered acceptable. This leads, usually, to the loss of the semantic relationship between the T2 FS and the concept it models. As a consequence, the interpretability of some of the ESs and the explainability of the uncertainty measures obtained from them can decrease. To overcome these issues, constrained type-2 (CT2) fuzzy sets have been proposed. However, no formal definitions for some of their key components [e.g., acceptable ESs (AESs)] and constrained operations have been given. In this article, we provide some theoretical underpinning for the definition of CT2 sets, their inferencing and defuzzification method. To conclude, the constrained inference framework is presented, applied to two real-world cases and briefly compared to the standard interval type-2 inference and defuzzification method.

Constrained interval type-2 (CIT2) fuzzy sets have been introduced to preserve interpretability when moving from type-1 to interval type-2 (IT2) membership functions. Although they can be used to produce type-2 fuzzy systems with enhanced explainability, so far, the latter comes at the expense of high computational cost. Specifically, the exhaustive type-reduction method for CIT2 Mamdani systems has been shown to be too slow to be used in practical applications and even the current approximation procedure is much slower than modern type-reduction algorithms used for IT2 fuzzy sets. In this article, a novel type-reduction procedure for CIT2 sets is presented, based on the concept of switch indices. The algorithm is applied on a real-world classification problem and compared to other type-reduction approaches used in IT2 and CIT2 systems. In the case studies presented, the new algorithm is significantly faster than the exhaustive and sampling CIT2 approaches while keeping the high level of interpretability of the type-reduction operation that characterizes CIT2 fuzzy sets.

The type-1 ordered weighted averaging (T1OWA) operator has demonstrated the capacity for directly aggregating multiple sources of linguistic information modeled by fuzzy sets rather than crisp values. Yager's ordered weighted averaging (OWA) operators possess the properties of idempotence, monotonicity, compensativeness, and commutativity. This article aims to address whether or not T1OWA operators possess these properties when the inputs and associated weights are fuzzy sets instead of crisp numbers. To this end, a partially ordered relation of fuzzy sets is defined based on the fuzzy maximum (join) and fuzzy minimum (meet) operators of fuzzy sets, and an alpha-equivalently-ordered relation of groups of fuzzy sets is proposed. Moreover, as the extension of orness and andness of an Yager's OWA operator, joinness and meetness of a T1OWA operator are formalized, respectively. Then, based on these concepts and the representation theorem of T1OWA operators, we prove that T1OWA operators hold the same properties as Yager's OWA operators possess, i.e., idempotence, monotonicity, compensativeness, and commutativity. Various numerical examples and a case study of diabetes diagnosis are provided to validate the theoretical analyses of these properties in aggregating multiple sources of uncertain information and improving integrated diagnosis, respectively.

In recent year, there has been a growing need for intelligent systems that not only are able to provide reliable classifications but can also produce explanations for the decisions they make. The demand for increased explainability has led to the emergence of explainable artificial intelligence (XAI) as a specific research field. In this context, fuzzy logic systems represent a promising tool thanks to their inherently interpretable structure. The use of a rule-base and linguistic terms, in fact, have allowed researchers to create models that are able to produce explanations in natural language for each of the classifications they make. So far, however, designing systems that make use of interval type-2 (IT2) fuzzy logic and also give explanations for their outputs has been very challenging, partially due to the presence of the type-reduction step. In this paper, it will be shown how constrained interval type-2 (CIT2) fuzzy sets represent a valid alternative to conventional interval type-2 sets in order to address this issue. Through the analysis of two case studies from the medical domain, it is shown how explainable CIT2 classifiers are produced. These systems can explain which rules contributed to the creation of each of the endpoints of the output interval centroid, while showing (in these examples) the same level of accuracy as their IT2 counterpart.

Constrained interval type-2 (CIT2) fuzzy sets are a class of type-2 fuzzy sets that has been recently proposed as a way to extend type-1 membership functions to interval type-2 (IT2) while keeping a semantic connection between the IT2 fuzzy set and the concept it models. Recent work has shown how their mathematical properties can be used to design CIT2 fuzzy logic systems that are able to provide explanations for their outputs. Although the CIT2 representation can be a valuable alternative to the IT2 one, no software library for their implementation is available for the research community. The aim of this paper is to introduce a new Java library, Juzzy Constrained, that has been developed as an extension of the popular type-1 and type-2 Java toolkit Juzzy, adding support for CIT2 sets and systems. Throughout the paper, the main classes and the structure of the new library are described, together with a working example that illustrates how to build a CIT2 fuzzy system from scratch and how it can be used to produce explanations for the output.

Many decision making processes are based on choosing options with maximum utility. Often utility assessments are associated with uncertainty, which may be mathematically modeled by intervals of utilities. Intervals of utilities may be mapped to single utility values by so-called type reduction methods which have been originally developed in the context of interval type-2 defuzzification: the method by Nie and Tan (NT), consistent linear type reduction (CLTR), consistent quadratic type reduction (CQTR), and the uncertainty weight method (UW). This paper considers the problem of comparing pairs of utility intervals using type reduction methods. Three different possible relations between pairs of intervals (disjoint, overlapping, and inclusive) are distinguished in an extensive experimental study, which yields recommendations for the choice of type reduction methods with respect to the level of risk that the decision maker is willing to take. If the focus is on mean utility, then we recommend the Nie-Tan method. For more cautious decision making, when very low utilities should be avoided, we recommend consistent linear type reduction with a high value of the cautiousness parameter or consistent quadratic type reduction. For more risky decision making with a strong focus on very high utilities we recommend consistent linear type reduction with a low value of the cautiousness parameter.

Alpha-cut representation of fuzzy sets has been used as a basis for fuzzy numbers ranking in some applications but rarely used for defuzzification of rule-based systems or fuzzy controllers . Moreover, such alpha-cut defuzzification (called ACD here) is not yet formally linked to the membership function (MF) or to the common MF-based defuzzification methods, namely the centroid . The ACD can be considered as a generalisation of the similar algorithms in fuzzy numbers to any fuzzy set. A close-form formula for ACD is developed that involves both MF and its derivative, which shows that ACD reflects both static and dynamic aspects of a fuzzy set. Moreover, formal links between ACD and some MF-based defuzzification methods are shown. Through two groups of experiments, the utility of the new method is compared with centroid defuzzification. Particularly, we examined how the ACD significantly outperforms the centroid for noisy time-series prediction. Finally, the computation complexity of ACD is shown to be about the same as the centroid method, for convex fuzzy sets . Our tests suggest that ACD can be considered as a viable alternative defuzzification method for fuzzy system designers.

This article provides new application-independent perspectives about the performance potential of an intuitionistic (I-) fuzzy system over a (classical) Takagi-Sugeno-Kang (TSK) fuzzy system. It does this by extending sculpting the state-space works from a TSK fuzzy system to an I-fuzzy system. It demonstrates that, for piecewise-linear membership functions (trapezoids and triangles), an I-fuzzy system always has significantly more first-order rule partitions of the state space-the coarse sculpting of the state space-than does a TSK fuzzy system, and that some I-fuzzy systems also have more second-order rule partitions of the state space-the fine sculpting of the state space-than does a TSK fuzzy system. It is the author's conjecture that for piecewise-linear membership functions (trapezoids and triangles): it is the always significantly greater coarse (and possibly fine) sculpting of the state space that provides an I-fuzzy system with the potential to outperform a TSK fuzzy system, and that a type-1 I-fuzzy system has the potential to outperform an interval type-2 fuzzy system.

Constrained type-2 fuzzy sets have been proposed as a tool to model type-2 fuzzy sets starting from a type-1 generator set with uncertainty. This constrained representation only defines as acceptable the embedded sets that have the same shape as the generator set in order to process only membership functions that are considered "meaningful" when using fuzzy operators such as centroid defuzzification. However, the idea of "meaningfulness" has never been clearly defined; at the same time there are some contexts in which a given concept (e.g. medium height) can be reasonably represented by multiple shapes, such as triangular and Gaussian. The aim of this paper is both to formally define the idea of meaningfulness of shapes and to extend the formal definitions of constrained interval type-2 fuzzy sets in order to allow the presence of multiple shapes among the acceptable embedded sets.

Citizen science is becoming mainstream in a wide variety of real-world applications in astronomy or bioinformatics, in which, for example, classification tasks by experts are very time consuming. These projects engage amateur volunteers that are tasked to manually classify unannotated examples. As a result, we obtain a larger volume of labelled data that, however, contains a great level of uncertainty due to the wide range of expertise of the volunteers. Handling that inherent uncertainty is key to building robust and fast machine learning models that maximise the outcome of citizen science projects. In this work, we introduce a preliminary approach that first transforms the original results from a citizen science project to handle the uncertainty, and then uses this as input to a fuzzy k-nearest neighbour classifier. We leverage citizen science results in such a way that it naturally speeds up the learning and classification phases of the fuzzy classifier, and improves the classification performance. As a case study, we will focus on the Galaxy Zoo project that consisted of galaxy image classification. Our experimental results show that an appropriate use of citizen science data enables a faster and more robust classification using the fuzzy k-nearest neighbour classifier.

We propose the application of interval type-2 fuzzy decision making (IT2FDM) to dynamic scheduling of deliveries in a just-in-time logistic process. Delivery decisions are based on order priorities computed from the expected decrease of customer satisfaction for each order. We compare IT2FDM with first in first out (FIFO), earliest due date first (EDDF), and (type-1) fuzzy decision making (FDM). In a simulation of a real world process for a duration of 600 days IT2FDM in comparison with the three other methods yields the highest just-in-time delivery rate, the highest average customer satisfaction, and the highest percentage of very satisfied customers. The increasing percentage of very satisfied customers at the same time leads to a slightly increasing percentage of not satisfied customers, but the trade- off between these percentages can be balanced by an appropriate choice of the risk parameter.

Hot spot identification problems are present across a wide range of areas, such as transportation, health care and energy. Hot spots are locations where a certain type of event occurs with high frequency. A recent big data approach is capable of identifying hot spots in a dynamic manner, through the processing of large volumes of sensor data arriving as a stream. However, the method may produce imprecise results due to its crisp interpretation of hot spot locations and reliance on a fixed hot spot radius value. This paper presents an initial approach to addressing this shortcoming through incorporating the concept of fuzzy hot spots into the process. Experimental results on large real-world transportation datasets demonstrate the improved way in which this approach handles uncertainty in the definition of hot spots, and highlight promising future research areas for further application of fuzzy systems to the hot spot identification problem.

Fuzzy systems mimic human reasoning and provide solutions to problems under uncertainty via `computing with words'. This particular strength of fuzzy systems is often discarded in some real world applications where the fuzzy sets are designed for control problems or created through training using historical data. This study explores the interpretability of fuzzy systems by generating `meaningful' fuzzy sets using a dictionary constructed by humans and fuzzy transfer learning. The inverted pendulum control problem is used as a case study. The empirical results show that intepretability of a fuzzy system is achievable even for this problem at the expense of a `slightly' reduced performance.

This paper describes the deployment of the Mobile Opportunistic Disconnection-Tolerant Networks and Systems platform in heterogeneous real-world scenarios. Our aim is to prototype and test fully-distributed multi-layer complex DTN forwarding and processing algorithms in different real-time dynamic topologies with varying mobility, network resources and data workload (publish-subscribe) patterns. In this work, we describe two different real-world contexts of mobile heterogeneous sensing and query in disconnection-prone environments: 1) networked drones and vehicles for agricultural monitoring applications and 2) vehicles and drone communications for smart city applications in urban environments. We discuss the performance characteristics of different mobile disconnection prone communication protocols and dynamic architectures across a range of criteria.

Metaheuristics, being tailored to each particular domain by experts, have been successfully applied to many computationally hard optimization problems. However, once implemented, their application to a new problem domain or a slight change in the problem description would often require additional expert intervention. There is a growing number of studies on reusable cross-domain search methodologies, such as selection hyper-heuristics, which are applicable to problem instances from various domains, requiring minimal expert intervention or even none. This paper introduces a new learning automata-based selection hyper-heuristic controlling a set of multiobjective metaheuristics. The approach operates above three well-known multiobjective evolutionary algorithms and mixes them, exploiting the strengths of each algorithm. The performance and behavior of two variants of the proposed selection hyper-heuristic, each utilizing a different initialization scheme are investigated across a range of unconstrained multiobjective mathematical benchmark functions from two different sets and the real-world problem of vehicle crashworthiness. The empirical results illustrate the effectiveness of our approach for cross-domain search, regardless of the initialization scheme, on those problems when compared to each individual multiobjective algorithm. Moreover, both variants perform significantly better than some previously proposed selection hyper-heuristics for multiobjective optimization, thus significantly enhancing the opportunities for improved multiobjective optimization.

Although driving behavior has been largely studied amongst private motor vehicles drivers, the literature addressing heavy goods vehicle (HGV) drivers is scarce. Identifying the existing groups of driving stereotypes and their proportions enables researchers, companies, and policy makers to establish group-specific strategies to improve safety and economy. In addition, insight into driving styles can help predict drivers' reactions and therefore enable the modeling of interactions between vehicles and the possible obstacles encountered on a journey. Consequently, there are also contributions to the research and development of autonomous vehicles and smart roads. In this paper, our interest lies in investigating driving behavior within the HGV community in the United Kingdom (U.K.). We conduct analysis of a telematics dataset containing the incident information on 21 193 HGV drivers across the U.K. We are interested in answering two research questions: 1) What groups of behavior are we able to uncover? 2) How do these groups complement current findings in the literature? To answer these questions, we apply a two-stage data analysis methodology involving consensus clustering and ensemble classification to the dataset. Through the analysis, eight patterns of behavior are uncovered. It is also observed that although our findings have similarities to those from previous work on driving behavior, further knowledge is obtained, such as extra patterns and driving traits arising from vehicle and road characteristics.

Smart networks and services leverage in-network caching to improve transmission efficiency and support large amount of content sharing, decrease high operating costs and handle disconnections. In this paper, we investigate complex challenges related to content popularity weighting process in collaborative caching algorithm in heterogeneous mobile disconnection prone environments. We describe a reputation-based popularity weighting mechanism built in information-centric layer of our adaptive collaborative Congestion-Aware Forwarding, Replication and Caching framework CafRepCache which considers a realistic case where caching points gathering content popularity observed by nodes differentiate between them according to nodes' reputation and network's connectivity values. We extensively evaluate CafRepCache against two competitive protocols across three heterogeneous real-world mobility, connectivity traces and use YouTube dataset for different workload and content popularity patterns. We show that CafRepCache balances the trade-offs to achieve higher cache hit ratio, efficiency and success ratios while keeping lower delays, packet loss and caching footprint compared to competing protocols across real-world traces in the face of dynamic mobility of publishers and subscribers.

The Karnik-Mendel algorithm is used to compute the centroid of interval type-2 fuzzy sets, determining the switch points needed for the lower and upper bounds of the centroid, through an iterative process. It is commonly acknowledged that there is no closed-form solution for determining such switch points. Many enhanced algorithms have been proposed to improve the computational efficiency of the Karnik-Mendel algorithm. However, all of these algorithms are still based on iterative procedures. In this paper, a direct approach based on derivatives for determining the switch points without multiple iterations has been proposed, together with mathematical proof that these switch points are correctly determining the lower and upper bounds of the centroid. Experimental simulations show that the direct approach obtains the same switch points, but is more computationally efficient than any of the existing (iterative) algorithms. Thus, we propose that this algorithm should be used in any application of interval type-2 fuzzy sets in which the centroid is required.

This letter is a supplement to the previous paper “A Direct Approach for Determining the Switch Points in the Karnik-Mendel Algorithm”. In the previous paper, the enhanced iterative algorithm with stop condition (EIASC) was shown to be the most inefficient in R. Such outcome is apparently different from the results in another paper in which EIASC was illustrated to be the most efficient in MATLAB. An investigation has been made into this apparent inconsistency and it can be confirmed that both the results in R and MATLAB are valid for the EIASC algorithm. The main reason for such phenomenon is the efficiency difference of loop operations in R and MATLAB. It should be noted that the efficiency of an algorithm is closely related to its implementation in practice. In this letter, we update the comparisons of the three algorithms in the previous paper, based on optimized implementations under five programming languages (MATLAB, R, Python, C, and Java). From this, we conclude that results in one programming language cannot be simply extended to all languages.

This paper presents an approach to prediction based on a new interval type-2 Atanassov-intuitionistic fuzzy logic system (IT2AIFLS) of the Takagi-Sugeno-Kang fuzzy inference with neural network learning capability. The gradient descent algorithm is used to adapt the parameters of the IT2AIFLS. The empirical comparison is made on the designed system using some benchmark regression problems-both artificial and real-world datasets. Analyses of our results reveal that IT2AIFLS outperforms its type-1 variant, other type-1 fuzzy logic approaches, and some type-2 fuzzy systems in the regression tasks. The reason for the improved performance of the proposed framework of IT2AIFLS is the introduction of nonmembership functions and intuitionistic fuzzy indices into the classical IT2FLS model. This increases the level of fuzziness in the proposed IT2AIFLS framework, thus providing more accurate approximations than AIFLS, classical type-1, and interval type-2 fuzzy logic systems.

This paper presents a novel application of a hybrid learning approach to the optimisation of membership and nonmembership functions of a newly developed interval type-2 intuitionistic fuzzy logic system (IT2 IFLS) of a Takagi-Sugeno-Kang (TSK) fuzzy inference system with neural network learning capability. The hybrid algorithms consisting of decoupled extended Kalman filter (DEKF) and gradient descent (GD) are used to tune the parameters of the IT2 IFLS for the first time. The DEKF is used to tune the consequent parameters in the forward pass while the GD method is used to tune the antecedents parts during the backward pass of the hybrid learning. The hybrid algorithm is described and evaluated, prediction and identification results together with the runtime are compared with similar existing studies in the literature. Performance comparison is made among the proposed hybrid learning model of IT2 IFLS, a TSK-type-1 intuitionistic fuzzy logic system (IFLS-TSK), and a TSK-type interval type-2 fuzzy logic system (IT2 FLS-TSK) on two instances of the datasets under investigation. The empirical comparison is made on the designed systems using three artificially generated datasets and three real world datasets. Analysis of results reveal that IT2 IFLS outperforms its type-1 variants, IT2 FLS and most of the existing models in the literature. Moreover, the minimal run time of the proposed hybrid learning model for IT2 IFLS also puts this model forward as a good candidate for application in real time systems.

Type-reduction of type-2 fuzzy sets is considered to be a defuzzification bottleneck because of the computational complexity involved in the process of type-reduction. In this paper, we prove that the closed-form Nie-Tan operator, which outputs the average of the upper and lower bounds of the footprint of uncertainty, is actually an accurate method for defuzzifying interval type-2 fuzzy sets.

Simulated Annealing is a well known local search metaheuristic used for solving computationally hard optimisation problems. Cross-domain search poses a higher level issue where a single solution method is used with minor, preferably no modification for solving characteristically different optimisation problems. The performance of a metaheuristic is often dependant on its initial parameter settings, hence detecting the best configuration, i.e. parameter tuning is crucial, which becomes a further challenge for cross-domain search. In this paper, we investigate the cross-domain search performance of Simulated Annealing via tuning for solving six problems, ranging from personnel scheduling to vehicle routing under a stochastic local search framework. The empirical results show that Simulated Annealing is extremely sensitive to the initial parameter settings leading to sub-standard performance when used as a single solution method for cross-domain search. Moreover, we demonstrate that cross-domain parameter tuning is inferior to domain-level tuning highlighting the requirements for adaptive parameter configurations when dealing with cross-domain search.

Multi-objective evolutionary algorithms (MOEAs) based on the concept of Pareto-dominance have been successfully applied to many real-world optimisation problems. Recently, research interest has shifted towards indicator-based methods to guide the search process towards a good set of trade-off solutions. One commonly used approach of this nature is the indicator-based evolutionary algorithm (IBEA). In this study, we highlight the solution distribution issues within IBEA and propose a modification of the original approach by embedding an additional Pareto-dominance based component for selection. The improved performance of the proposed modified IBEA (mIBEA) is empirically demonstrated on the well-known DTLZ set of benchmark functions. Our results show that mIBEA achieves comparable or better hypervolume indicator values and epsilon approximation values in the vast majority of our cases (13 out of 14 under the same default settings) on DTLZ1-7. The modification also results in an over 8-fold speed-up for larger populations.

A selection hyper-heuristic is a search method that controls a prefixed set of low-level heuristics for solving a given computationally difficult problem. This study investigates a learning-via demonstrations approach generating a selection hyper-heuristic for Open Vehicle Routing Problem (OVRP). As a chosen `expert' hyper-heuristic is run on a small set of training problem instances, data is collected to learn from the expert regarding how to decide which low-level heuristic to select and apply to the solution in hand during the search process. In this study, a Time Delay Neural Network (TDNN) is used to extract hidden patterns within the collected data in the form of a classifier, i.e an `apprentice' hyper-heuristic, which is then used to solve the `unseen' problem instances. Firstly, the parameters of TDNN are tuned using Taguchi orthogonal array as a design of experiments method. Then the influence of extending and enriching the information collected from the expert and fed into TDNN is explored on the behaviour of the generated apprentice hyper-heuristic. The empirical results show that the use of distance between solutions as an additional information collected from the expert generates an apprentice which outperforms the expert algorithm on a benchmark of OVRP instances.

In a previous paper, we proposed an extended ANFIS architecture and showed that interval type-2 ANFIS produced larger errors than type-1 ANFIS on the well-known IRIS classification problem. In this paper, more experiments on both synthetic and real-world data are conducted to further investigate and compare the performance of interval type-2 ANFIS and type-1 ANFIS. For each dataset, interval type-2 ANFIS is optimised in three different ways, including a strategy suggested by Mendel such that interval type-2 ANFIS would be no worse than type-1 ANFIS. Our results show that in some circumstances the performance of interval type-2 ANFIS can be improved when it is initialised with blurred optimised type-1 ANFIS parameters. However, in general, interval type-2 ANFIS does not produce a clear performance improvement compared to type-1 ANFIS, especially on Mackey-Glass data with large noise. Thus, we conclude that the choice of interval type-2 ANFIS over type-1 ANFIS should be carefully considered, since type-2 ANFIS is more computationally complex, yet significantly better performance cannot be easily obtained.

Conventional fuzzy time series approaches make use of type-1 or type-2 fuzzy models. Type-1 models with one index (membership grade) cannot fully handle the level of uncertainty inherent in many real world applications. The type-2 models with upper and lower membership functions do handle uncertainties in many applications better than its type-1 counterparts. This study proposes the use of interval type-2 intuitionistic fuzzy logic system of Takagi-Sugeno-Kang (IT2IFLS-TSK) fuzzy inference that utilises more parameters than type-2 fuzzy models in time series forecasting. The IT2IFLS utilises more indexes namely upper and lower non-membership functions. These additional parameters of IT2IFLS serve to refine the fuzzy relationships obtained from type-2 fuzzy models and ultimately improve the forecasting performance. Evaluation is made on the proposed system using three real world benchmark time series problems namely: Santa Fe, tree ring and Canadian lynx datasets. The empirical analyses show improvements of prediction of IT2IFLS over other approaches on these datasets.

As non-singleton fuzzy logic controllers (NSFLCs) are capable of capturing input uncertainties, they have been effectively used to control and navigate unmanned aerial vehicles (UAVs) recently. To further enhance the capability to handle the input uncertainty for the UAV applications, a novel NSFLC with the recently introduced similarity-based inference engine, i.e., Sim-NSFLC, is developed. In this paper, a comparative study in a 3D trajectory tracking application has been carried out using the aforementioned Sim-NSFLC and the NSFLCs with the standard as well as centroid composition-based inference engines, i.e., Sta-NSFLC and Cen-NSFLC. All the NSFLCs are developed within the robot operating system (ROS) using the C++ programming language. Extensive ROS Gazebo simulation-based experiments show that the Sim-NSFLCs can achieve better control performance for the UAVs in comparison with the Sta-NSFLCs and Cen-NSFLCs under different input noise levels.

In this paper, our aim is to compare and contrast various ways of modeling uncertainty by using different type-2 fuzzy membership functions available in literature. In particular we focus on a novel type-2 fuzzy membership function, - “Elliptic membership function”. After briefly explaining the motivation behind the suggestion of the elliptic membership function, we analyse the uncertainty distribution along its support, and we compare its uncertainty modeling capability with the existing membership functions. We also show how the elliptic membership functions perform in fuzzy arithmetic. In addition to its extra advantages over the existing type-2 fuzzy membership functions such as having decoupled parameters for its support and width, this novel membership function has some similar features to the Gaussian and triangular membership functions in addition and multiplication operations. Finally, we have tested the prediction capability of elliptic membership functions using interval type-2 fuzzy logic systems on US Dollar/Euro exchange rate prediction problem. 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 membership functions have comparable prediction results when compared to Gaussian and triangular membership functions.

Non-singleton fuzzification is used to model uncertain (e.g. noisy) inputs within fuzzy logic systems. In the standard approach, assuming the fuzzification type is known, the observed [noisy] input is usually considered to be the core of the input fuzzy set, usually being the centre of its membership function. This paper proposes a new fuzzification method (not type), in which the core of an input fuzzy set is not necessarily located at the observed input, rather it is dynamically adjusted based on statistical methods. Using the weighted moving average, a few past samples are aggregated to roughly estimate where the input fuzzy set should be located. While the added complexity is not huge, applying this method to the well-known Mackey-Glass and Lorenz time-series prediction problems, show significant error reduction when the input is corrupted by different noise levels.

Fuzzy logic systems have been extensively applied for solving many real world application problems because they are found to be universal approximators and many methods, particularly, gradient descent (GD) methods have been widely adopted for the optimization of fuzzy membership functions. Despite its popularity, GD still suffers some drawbacks in terms of its slow learning and convergence. In this study, the use of decoupled extended Kalman filter (DEKF) to optimize the parameters of an interval type-2 intuitionistic fuzzy logic system of Tagagi-Sugeno-Kang (IT2IFLS-TSK) fuzzy inference is proposed and results compared with IT2IFLS gradient descent learning. The resulting systems are evaluated on a real world dataset from Australia's electricity market. The IT2IFLS-DEKF is also compared with its type-1 variant and interval type-2 fuzzy logic system (IT2FLS). Analysis of results reveal performance superiority of IT2IFLS trained with DEKF (IT2IFLS-DEKF) over IT2IFLS trained with gradient descent (IT2IFLS-GD). The proposed IT2IFLS-DEKF also outperforms its type-1 variant and IT2FLS on the same learning platform.

Supplier selection and inventory planning are critical and challenging tasks in Supply Chain Management. There are many studies on both topics and many solution techniques have been proposed dealing with each problem separately. In this study, we present a two-stage integrated approach to the supplier selection and inventory planning. In the first stage, suppliers are ranked based on various criteria, including cost, delivery, service and product quality using Interval Type-2 Fuzzy Sets (IT2FS)s. In the following stage, an inventory model is created. Then, an Multi-objective Evolutionary Algorithm (MOEA) is utilised simultaneously minimising the conflicting objectives of supply chain operation cost and supplier risk. We evaluated the performance of three MOEAs with tuned parameter settings, namely NSGA-II, SPEA2 and IBEA on a total of twenty four synthetic and real world problem instances. The empirical results show that in the overall, NSGA-II is the best performing MOEA producing high quality trade-off solutions to the integrated problem of supplier selection and inventory planning.

Type-2 fuzzy logic systems make use of type-2 fuzzy sets (T2FS). To be able to deliver useful type-2 fuzzy logic applications we need to be able to perform meaningful operations on these sets. These operations should also be practically tractable. However, T2FS suffer the shortcoming of being complex by definition. Indeed, the third dimension, which is the source of extra parameters, is in itself the origin of extra computational cost. The quest for a representation that allow practical systems to be implemented is the motivation for our work. In this paper, we define the alpha-cut decomposition theorem for T2FS which is a new representation analogous to the alpha-cut representation of type-1 fuzzy sets and the extension principle. We show that this new decomposition theorem forms a methodology for extending mathematical concepts from crisp sets to T2FS directly. In the process of developing this theory, we also define a generalization that allows us to extend operations from interval T2FS or interval valued fuzzy sets to T2FS. These results will allow for the more applications of T2FS by expiating the parallelism that the research here affords.

There is a growing number of studies on general purpose metaheuristics that are directly applicable to multiple domains. Parameter setting is a particular issue considering that many of such search methods come with a set of parameters to be configured. Fuzzy logic has been used extensively in control applications and is known for its ability to handle uncertainty. In this study, we investigate the potential of using fuzzy systems to control the parameter settings of a threshold accepting (TA) metaheuristic for improving the overall effectiveness of a cross-domain approach. We have evaluated the performance of various general purpose local search metaheuristics which mix multiple heuristics at random and apply the TA metaheuristic with fixed threshold, crisp (non-fuzzy) rule-based control of the threshold and various fuzzy systems controlling the threshold. The empirical results show that the approach using the TA with crisp rule-based control performs the best across six problem domains from a benchmark.

In this paper, an extended ANFIS architecture is proposed. By incorporating an extra layer for the fuzzification process, the extended architecture is able to fit both type-1 and interval type-2 models. The learning properties of the proposed architecture based on the least-squares estimate method are studied on selected type-1 and interval type-2 ANFIS models. We show that the least-squares estimate method in general behaves differently for interval type-2 ANFIS models compared to type-1 ANFIS models, producing larger errors for interval type-2 ANFIS.

Fuzzy logic controllers (FLCs) have extensively been used for the autonomous control and guidance of unmanned aerial vehicles (UAVs) due to their capability of handling uncertainties and delivering adequate control without the need for a precise, mathematical system model which is often either unavailable or highly costly to develop. Despite the fact that non-singleton FLCs (NSFLCs) have shown more promising performance in several applications when compared to their singleton counterparts (SFLCs), most of UAV applications are still realized by using SFLCs. In this paper, we explore the potential of both standard and the recently introduced centroid based NSFLCs, i.e., Sta-NSFLC and Cen-NSFLC, for the control of a quadcopter UAV under various input noise conditions using different levels of fuzzifier, and a comparative study has been conducted using the three aforementioned FLCs. We present a series of simulation-based experiments, the simulation results show that the control performances of NSFLCs are better than those of SFLC, and the Cen-NSFLC outperforms the Sta-NSFLC especially under highly noisy conditions.

In non-singleton fuzzy logic systems (NSFLSs) input uncertainties are modelled with input fuzzy sets in order to capture input uncertainty such as sensor noise. The performance of NSFLSs in handling such uncertainties depends both on the actual input fuzzy sets (and their inherent model of uncertainty) and on the way that they affect the inference process. This paper proposes a novel type of NSFLS by replacing the composition-based inference method of type-1 fuzzy relations with a similarity-based inference method that makes NSFLSs more sensitive to changes in the input's uncertainty characteristics. The proposed approach is based on using the Jaccard ratio to measure the similarity between input and antecedent fuzzy sets, then using the measured similarity to determine the firing strength of each individual fuzzy rule. The standard and novel approaches to NSFLSs are experimentally compared for the well-known problem of Mackey-Glass time series predictions, where the NSFLS's inputs have been perturbed with different levels of Gaussian noise. The experiments are repeated for system training under both noisy and noise-free conditions. Analyses of the results show that the new method outperforms the standard approach by substantially reducing the prediction errors.

This paper presents an approach to prediction based on a new interval type-2 intuitionistic fuzzy logic system (IT2IFLS) of Takagi-Sugeno-Kang (TSK) fuzzy inference. The gradient descent algorithm (GDA) is used to adapt the parameters of the IT2IFLS. The empirical comparison is made on the designed system using two synthetic datasets. Analysis of our results reveal that the presence of additional degrees of freedom in terms of non-membership functions and hesitation indexes in IT2IFLS tend to reduce the root mean square error (RMSE) of the system compared to a type-1 fuzzy logic approach and some interval type-2 fuzzy systems.

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.

The main aim of this paper is to connect R-fuzzy sets and type-2 fuzzy sets, so as to provide a practical means to express complex uncertainty without the associated difficulty of a type-2 fuzzy set. The paper puts forward a significance measure, to provide a means for understanding the importance of the membership values contained within an R-fuzzy set. The pairing of an R-fuzzy set and the significance measure allows for an intermediary approach to that of a type-2 fuzzy set. By inspecting the returned significance degree of a particular membership value, one is able to ascertain its true significance in relation, relative to other encapsulated membership values. An R-fuzzy set coupled with the proposed significance measure allows for a type-2 fuzzy equivalence, an intermediary, all the while retaining the underlying sentiment of individual and general perspectives, and with the adage of a significantly reduced computational burden. Several human based perception examples are presented, wherein the significance degree is implemented, from which a higher level of detail can be garnered. The results demonstrate that the proposed research method combines the high capacity in uncertainty representation of type-2 fuzzy sets, together with the simplicity and objectiveness of type-1 fuzzy sets. This in turn provides a practical means for problem domains where a type-2 fuzzy set is preferred but difficult to construct due to the subjective type-2 fuzzy membership.

This paper reports the use of simulated annealing to design more efficient fuzzy logic systems to model problems with associated uncertainties. Simulated annealing is used within this work as a method for learning the best configurations of interval and general type-2 fuzzy logic systems to maximize their modeling ability. The combination of simulated annealing with these models is presented in the modeling of four benchmark problems including real-world problems. The type-2 fuzzy logic system models are compared in their ability to model uncertainties associated with these problems. Issues related to this combination between simulated annealing and fuzzy logic systems, including type-2 fuzzy logic systems, are discussed. The results demonstrate that learning the third dimension in type-2 fuzzy sets with a deterministic defuzzifier can add more capability to modeling than interval type-2 fuzzy logic systems. This finding can be seen as an important advance in type-2 fuzzy logic systems research and should increase the level of interest in the modeling applications of general type-2 fuzzy logic systems, despite their greater computational load.

Stability analysis is an important research direction in evolutionary game theory. Evolutionarily stable states have a close relationship with Nash equilibria of repeated games, which are characterized by the folk theorem. When applying the folk theorem, one needs to compute the minimax profile of the game in order to find Nash equilibria. Computing the minimax profile is an NP-hard problem. In this paper, we investigate a new methodology to compute evolutionary stable states based on the level-k equilibrium, a new refinement of Nash equilibrium in repeated games. A level-k equilibrium is implemented by a group of players who adopt reactive strategies and who have no incentive to deviate from their strategies simultaneously. Computing the level-k equilibria is tractable because the minimax payoffs and strategies are not needed. As an application, this paper develops a tractable algorithm to compute the evolutionarily stable states and the Pareto front of n-player symmetric games. Three games, including the iterated prisoner's dilemma, are analyzed by means of the proposed methodology.

Selection of an appropriate supplier is a crucial and challenging task in the effective management of a supply chain. Also, appropriate inventory management is critical to the success of a supply chain operation. In recent years, there has been a growing interest in the area of selection of an appropriate vendor and creating good inventory planning using supplier selection information. In this paper, we consider both of these tasks in a two-stage approach employing Interval Type-2 Fuzzy Sets (IT2FS) and Simulated Annealing (SA). In the first stage, the supplier selection problem is solved by using IT2FS for ranking the suppliers. We present an inventory model incorporating information from the first stage that captures the influence of supplier risk on the total cost of supply chain operation. In the second stage, SA is used for solving the inventory planning problem based on this model improving on both supply chain operation cost and supplier risk. In this study, we evaluated our approach using different scenarios and scalarisation techniques for SA to handle two objectives, simultaneously.

In this paper we investigate a multi-objective portfolio selection model with three criteria: risk, return and liquidity for investors. Non-probabilistic uncertainty factors in the market, such as imprecision and vagueness of investors’ preference and judgement are simulated in the portfolio selection process. The liquidity of portfolio cannot be accurately predicted in the market, and thus is measured by fuzzy set theory. Invertors’ individual preference and judgement are cooperated in the decision making process by using satisfaction functions to measure the objectives. A compromise based goal programming approach is applied to find compromised solutions. By this approach, not only can we obtain quality solutions in a reasonable computational time, but also we can achieve a trade-off between the objectives according to investors’ preference and judgement to enable a better decision making. We analyse the portfolio strategies obtained by using the proposed simulation approach subject to different settings in the satisfaction functions.

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.

Interval type-2 defuzzification maps an interval type-2 fuzzy set to a crisp number. We show that the semantic meaning of the interval type-2 fuzzy set (the associated opportunity or risk) has to be considered in the choice of an appropriate interval type-2 defuzzification method. Motivated by a list of “axioms” for type-1 defuzzification we introduce twelve mathematical properties for interval type-2 defuzzification that serve as a theoretical framework to assess different interval type-2 defuzzification methods. We show that the well-known Karnik-Mendel algorithm violates at least four of these twelve properties.

This paper reports on a new approach for automatic learning of general type-2 fuzzy logic systems (GT2FLSs) using simulated annealing (SA). The learning process in this work starts without an initial interval type-2 fuzzy system and has an objective to optimize all membership function parameters involved in the general type-2 fuzzy set in two stages. This is a novel methodology for learning GT2FLSs using the vertical-slices representation. The methodology used here is based on a proposed parameterization method presented in a previous work to ease the design of GT2FLSs. Two models of GT2FLSs have been applied using two different type-reduction techniques. The first technique is the sampling method, which is non-deterministic. The second technique is the vertical-slices centroid type-reduction (VSCTR), which is deterministic. Both models as well as an interval type-2 fuzzy logic system (IT2FLS) model have been applied to predict a Mackey-Glass time series. A comparison of the results of modeling these problems using the three models showed more accurate modeling for the GT2FLSs when using the VSCTR deterministic defuzzification method. It has also been shown that a GT2FLS with VSCTR defuzzification is more able to handle uncertainty than an IT2FLS, although the latter was faster.

This paper presents a new defuzzification algorithm for interval type-2 fuzzy sets. The algorithm exploits the fact that we can treat an interval type-2 fuzzy set as two type-2 fuzzy sets. We suggest in this paper that monotonicity is an important property for defuzzifiers and so we provide a definition of monotonicity for type-2 defuzzifiers based on previous work by Runkler. The research reported here shows that our new operator is monotonic and provides a defuzzified value that lies within the interval computed by the popular Karnik-Mendel algorithm.

In this paper, a new fuzzy regression model that is supported by support vector regression is presented. Type-2 fuzzy systems are able to tackle applications that have significant uncertainty. However general type-2 fuzzy systems are more complex than type-1 fuzzy systems. Support vector machines are similar to fuzzy systems in that they can also model systems that are non-linear in nature. In the proposed model the consequent parameters of type-2 fuzzy rules are learnt using support vector regression and an efficient closed-form type reduction strategy is used to simplify the computations. Support vector regression improved the generalisation performance of the fuzzy rule-based system in which the fuzzy rules were a set of interpretable IF-THEN rules. The performance of the proposed model was demonstrated by conducting case studies for the non-linear system approximation and prediction of chaotic time series. The model yielded promising results and the simulation results are compared to the results published in the area.

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.

In the literature, there is a presumption that a grey set and an interval-valued fuzzy set are equivalent. This presumption ignores the existence of discrete components in a grey number. In this paper, new measurements of uncertainties of grey numbers and grey sets, consisting of both absolute and relative uncertainties, are defined to give a comprehensive representation of uncertainties in a grey number and a grey set. Some simple examples are provided to illustrate that the proposed uncertainty measurement can give an effective representation of both absolute and relative uncertainties in a grey number and a grey set. The relationships between grey sets and interval-valued fuzzy sets are also analyzed from the point of view of the proposed uncertainty representation. The analysis demonstrates that grey sets and interval-valued fuzzy sets provide different but overlapping models for uncertainty representation in sets.

Software design patterns describe recurring design problems and provide the essence of best practice solutions. It is useful and important, for various software engineering tasks, to know which design pattern is implemented where in a software design. However, this information is often lost due to poor or absent documentation, and so accurate recognition tools are required. The problem is that design patterns, given their abstract and vague nature, have a level of resistance to be automatically and accurately recognized. Although this vagueness or fuzziness can be captured and modelled by the fuzzy inference system, it has not yet been applied to solve this problem. This paper fills this gap by proposing an approach for design pattern recognition based on Adaptive Neuro Fuzzy Inference System. Our approach consists of two phases: space reduction phase and design pattern recognition phase. Both phases are implemented by ANFIS. We evaluate the approach by an experiment conducted to recognize six design patterns in an open source application. The results show that the approach is viable and promising.

Grey system theory has many facets, one of which is the so-called GM(1,1) model, used for predicting and forecasting. This paper proposes a novel way of improving the overall relative accuracy of the new-information grey model, and the metabolic grey model, and also by improving the filter value accuracy. By incorporating a weight sequence that is populated by a genetic algorithm to minimize the error of the simulated values. The least square parameters (-a) and b, can then be scaled by the values contained in the weight sequence, until a satisfactory result is obtained. If a high level of accuracy can be attained for the simulation values of the model, and also for the filter value, it will ultimately allow for greater forecasting ability.

In this paper, simulated annealing algorithm is used to design general type-2 fuzzy logic systems (GT2FLS) with the aid of interval type-2 fuzzy logic systems (IT2FLS). The proposed practical design methodology aims to reduce computations needed to get the best footprint of uncertainty (FOU) using IT2FLS. Simulated annealing is used to learn IT2FLS followed by learning the secondary membership functions in GT2FLS using a novel parameterization method. This methodology has been applied to four bench-mark problems. some practical issues have been discussed and some analysis of the results obtained by IT2FLS and GT2FLS has been drawn.

In previous work the Authors defined the alpha-cut representation for type-2 fuzzy sets. The strength of this representation is that it allows type-2 fuzzy sets to be fully defined by a collection of crisp sets. Each of these crisp sets may be independently processed within a fuzzy logic system prior to defuzzification. This independence means this representation is ideal for parallel implementation by minimizing the requirement for iteration and synchronisation. In this work we investigate the parallel implementation of type-2 fuzzy logic operations on a massively parallel graphics processor unit. We explore the performance characteristics of this parallel implementation and contrast with a serial implementation in a series of experiments.

Numerous similarity measures between intuitionistic fuzzy sets are proposed in literature using different approaches. In this paper, relationship between some existing intuitionistic fuzzy similarity and distance measures are investigated. These relations are paramount for the choice of a similarity or a distance measure and its application for any research topic.

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.

In this paper, a combination of a Takagi-Sugeno fuzzy system (TSK) and simulated annealing is used to predict well known time series by searching for the best configuration of the fuzzy system. Simulated annealing is used to optimise the parameters of the antecedent and the consequent parts of the fuzzy system rules. The results of the proposed method are encouraging indicating that simulated annealing and fuzzy logic are able to combine well in time series prediction.

In this paper two contrasting approaches to the defuzzification of generalised type-2 fuzzy sets are investigated. The sampling method is a direct approach, whereas the α-planes method extends interval defuzzification to generalised type-2 sets, and therefore has to be used in conjunction with an interval technique such as the Karnik-Mendel Iterative Procedure. Experimental evaluations are made of the sampling and α-planes methods, with respect to both efficiency and accuracy, showing both strategies to be efficient and good approximations.

Alpha-cuts and the extension principle form a methodology for extending mathematical concepts from crisp sets to fuzzy sets. They have been applied to many operations, and have also been extended to interval valued fuzzy sets. Recently, some researchers defined new representations of type-2 fuzzy sets, namely, the alpha-plane representation and the zSlice representation. In this paper we investigate an alternative representation utilising alpha-cuts and the extension principle for interval valued fuzzy sets and type-2 fuzzy sets.

This research proposes a new model for constructing decision trees using interval-valued fuzzy membership values employing on look-ahead based fuzzy decision tree induction and interval-valued fuzzy sets. Most existing fuzzy decision trees do not consider the uncertainty associated with their membership values. However, precise values of fuzzy membership values are not always possible. In this paper, we represent fuzzy membership values as intervals to model uncertainty and employ the look-ahead based fuzzy decision tree induction method and Hamming distance of interval-valued fuzzy sets to construct decision trees. An example is given to demonstrate the effectiveness of the approach.

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.

Yager's ordered weighted averaging (OWA) operator has been widely used in soft decision making to aggregate experts' individual opinions or preferences for achieving an overall decision. The traditional Yager's OWA operator focuses exclusively on the aggregation of crisp numbers. However, human experts usually tend to express their opinions or preferences in a very natural way via linguistic terms. Type-2 fuzzy sets provide an efficient way of knowledge representation for modeling linguistic terms. In order to aggregate linguistic opinions via OWA mechanism, we propose a new type of OWA operator, termed type-2 OWA operator, to aggregate the linguistic opinions or preferences in human decision making modeled by type-2 fuzzy sets. A Direct Approach to aggregating interval type-2 fuzzy sets by type-2 OWA operator is suggested in this paper. Some examples are provided to delineate the proposed technique. © 2010 Wiley Periodicals, Inc.

We begin by surveying the available strategies for type-reducing a discretised type-2 fuzzy set to a type-1 fuzzy set, namely the exhaustive method, the Karnik-Mendel Iterative Procedure, the sampling method, the Greenfield-Chiclana Collapsing Defuzzifier and the Nie-Tan method. We go on to investigate mathematically what happens to the Representative Embedded Set Approximation as the domain discretisation becomes finer. This leads into a discussion of the relationship between the collapsing and Nie-Tan methods. An experimental comparison is made between the collapsing and Nie-Tan methods, with respect to both efficiency and accuracy.

Recommending appropriate follow-up treatment options to patients after diagnosis and primary (usually surgical) treatment of breast cancer is a complex decision making problem. Often, the decision is reached by consensus from a multi-disciplinary team of oncologists, radiologists, surgeons and pathologists. Non-stationary fuzzy sets have been proposed as a mechanism to represent and reason with the knowledge of such multiple experts. In this paper, we briefly describe the creation of a non-stationary fuzzy inference system to provide decision support in this context, and examine a number of alternative methods for interpreting the output of such a non-stationary inference system. The alternative interpretation methodologies and the experiments carried out to compare these methods are detailed. Results are presented which shown that using majority voting ensemble decision making from a non-stationary fuzzy system improves accuracy of the decision making. We conclude that non-stationary systems coupled with ensemble interpretation methods are worthy of further exploration.

In this paper, a novel type-1 OWA based non-stationary fuzzy system is proposed, in which the type-1 OWA operator is used in the fuzzy inference process to aggregate the non-stationary fuzzy outputs. The advantage of non-stationary fuzzy sets lies in their ability to model expert's variations in automated decision support systems. The proposed scheme offers an opportunity to combine different uncertain objects with uncertain weights into an overall decision in the fuzzy inference process. The agreement achieved between the proposed fuzzy system and clinical expert decision in selecting optimal treatment plans is used to evaluate the performance of the method. The experimental results on post-operative breast cancer treatments have demonstrated that the proposed fuzzy system can effectively diagnose breast cancer treatment in decision supports.

Type-2 fuzzy systems are increasing in popularity, and there are many examples of successful applications. While many techniques have been proposed for creating parsimonious type-1 fuzzy systems, there is a lack of such techniques for type-2 systems. The essential problem is to reduce the number of rules, while maintaining the system's approximation performance. In this paper, four novel indexes for ranking the relative contribution of type-2 fuzzy rules are proposed, which are termedR-values,c-values, omega1-values, and omega2-values. TheR-values of type-2 fuzzy rules are obtained by applying a QR decomposition pivoting algorithm to the firing strength matrices of the trained fuzzy model. Thec-values rank rules based on the effects of rule consequents, while the omega1-values and omega2-values consider both the rule-base structure (via firing strength matrices) and the output contribution of fuzzy rule consequents. Two procedures for utilizing these indexes in fuzzy rule selection (termed ldquoforward selectionrdquo and ldquobackward eliminationrdquo) are described. Experiments are presented which demonstrate that by using the proposed methodology, the most influential type-2 fuzzy rules can be effectively retained in order to construct parsimonious type-2 fuzzy models.

This paper presents a new efficient multi-objective evolutionary algorithm for solving computationally-intensive optimization problems. To support a high degree of parallelism, the algorithm is based on a steady-state design. For improved efficiency the algorithm utilizes a surrogate to identify promising candidate solutions and filter out poor ones. To handle the uncertainties associated with the approximative surrogate evaluations, a new method for multi-objective optimization is described which is generally applicable to all surrogate techniques. In this method, basically, surrogate objective values assigned to offspring are adjusted to consider the error of the surrogate. The algorithm is evaluated on the ZDT benchmark functions and on a real-world problem of manufacturing optimization. In assessing the performance of the algorithm, a new performance metric is suggested that combines convergence and diversity into one single measure. Results from both the benchmark experiments and the real-world test case indicate the potential of the proposed algorithm.

Developing a treatment plan for breast cancer patient is a very complex process. In this paper, we propose a scheme of inducing fuzzy rules that characterise breast caner treatment knowledge from data. These fuzzy rules can augment the human experts in the process of medical diagnosis to select optimal treatment for patients. The proposed machine learning scheme applies the particle swarm optimisation technique (PSO) to the construction of an optimal support vector machine (SVM) model for the sake of inducing accurate and parsimonious fuzzy rules and simultaneously reducing input space dimensions, in which a new fittness function that regularises the importance ranks of features with misclassification rate is suggested. The SVM-based fuzzy classifier evades the curse of dimensionality in high-dimensional breast cancer data space in the sense that the number of support vectors, which equals the number of induced fuzzy rules, is not related to the dimensionality. The experiments have shown that not only the classification performance achieved by the proposed fuzzy classifier outperforms the ones achieved by other methods in the literature, but also the input space dimension has been reduced greatly.

This paper defines a new parameter for describing the uncertainty of rough sets. Different from the roughness of a rough set, a global roughness measures the uncertainty of rough sets with respect to the entire information system. This is essential especially for a special rough set - boundary rough sets.We give the definition of global roughness of approximation and boundary rough sets, and analyse their properties. Some examples are also provided to show the complementary features of global roughness and roughness of rough sets.

Yager’s ordered weighted averaging (OWA) operator has been widely used in soft decision making to aggregate experts’ individual opinions or preferences for achieving an overall decision. The traditional Yager’s OWA operator focuses exclusively on the aggregation of crisp numbers. However, human experts usually tend to express their opinions or preferences in a very natural way via linguistic terms, like “important” , “very important”, “good” etc.. Type-2 fuzzy sets provide an efficient way of knowledge representation for modelling linguistic terms. In order to aggregate linguistic opinions via the OWA mechanism, we propose a new type of OWA operator, termed type-2 OWA operator that is able to aggregate type-2 fuzzy sets, and therefore to aggregate the linguistic opinions or preferences in human decision making. The necessary equations for performing type-2 OWA operations on aggregating interval type-2 fuzzy sets are derived in this paper. Some examples are provided to illustrate the proposed technique.

Generalized type-2 fuzzy logic systems cannot currently be used for practical problems because the amount of computation required to defuzzify a generalized type-2 fuzzy set is too large. This paper presents a new method for defuzzifing a type-2 fuzzy set. The new much faster technique is based on geometric representations and operations. The results of a real world example contained in this paper show this new approach to be over 200,000 times faster than type-reduction. We present a new method for assessing the accuracy of the membership function of a type-2 fuzzy set. This method is used to show that the new representation used by the defuzzifier is not detrimental to the accuracy of the set. We also discuss the differences between the new approach and type-reduction, identifying the origin of this massive improvement in execution speed.

Geometric Type-2 fuzzy sets allow us to model type-2 fuzzy sets using geometry providing significant improvement in speed of computation. This paper explores in detail how well the geometric approach models continuous type-2 fuzzy sets from an accuracy perspective. A new methodology is introduced for conducting the comparison. This methodology is used on some example sets and the performance of the geometric model is given. It shows that, in broad terms, geometric type-2 fuzzy sets are more accurate than the discrete equivalent.

The inferencing stage of a type-2 fuzzy inferencing system is driven by join and meet operations. As conventionally implemented these algorithms are computationally complex. This article introduces optimised implementations for these operations. These alternative procedures pre-suppose that the grid method of discretisation is adopted. Time comparisons between the alternative and conventional implementations have been undertaken, for join and meet operations coded firstly in isolation, and secondly, within a full prototype type-2 fuzzy inferencing system. The experimental results show that the optimisation affords marked time reduction, particularly under finer discretisation.

Sailing experts can explain basic sailing skills by rules about how to steer sails and rudder according to direction of target and wind. This paper describes how to transform the sailor's knowledge into Mamdani type fuzzy inference systems. The proposed system controls two actuators - rudder and sails - even during tack and jibe. In combination with an automatic weather routeing system the sailboat is able to reach any target completely autonomously. Experiments on a demonstration sailboat have shown excellent results. Detailed log data analysis shows manoeuvres carried out as expected by sailing experts.

In this paper, we propose two novel indices for type-2 fuzzy rule ranking to identify the most influential fuzzy rules in designing type-2 fuzzy logic systems, and name them as R-values and c-values of fuzzy rules separately. The R-values of type-2 fuzzy rules are obtained by applying QR decomposition in which there is no need to estimate a rank as required in the SVD-QR with column pivoting algorithm. The c-values of type-2 fuzzy rules are suggested to rank rules based on the effects of rule consequents. Experimental results on a signal recovery problem have shown that by using the proposed indices the most influential type-2 fuzzy rules can be effectively selected to construct parsimonious type-2 fuzzy models while the system performances are kept at a satisfied level.

This paper presents a novel approach to the representation of type-1 and type-2 fuzzy sets utilising computational geometry. To achieve this our approach borrows ideas from the field of computational geometry and applies these techniques in the novel setting of fuzzy logic. We provide new algorithms for various operations on type-1 and type-2 fuzzy sets and for defuzzification. Results of experiments indicate that this approach reduces the execution speed of these operations

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 type-reduction to arrive at a 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 the 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. We found that the non-stationary approach provided interesting results.

In this paper we introduce embedded type-2 fuzzy relations and provide an alternative proof of the extended sup-star composition that makes use of these embedded type-2 fuzzy relations. The paper commences by giving a background to some terms and definitions used in type-2 fuzzy logic. Definitions are given for a type-2 fuzzy set, a secondary membership function and an embedded set. Additionally the representation theorem is described along with an explanation of type-2 fuzzy relations. These preliminary sections provide the basis for the remainder of the paper, which introduces the concept of embedded fuzzy relations and makes use of this in order to provide an alternative proof of the extended sup-star composition.

Roughness is an important indicator for the uncertainty of a rough set. This paper analyses the roughness bounds for set-oriented rough set operations. A bound of the roughness of the union between two set-oriented rough sets could be determined by the roughness of the two operand set-oriented sets. In most cases, a bound could also be found for a difference set between two set-oriented rough sets. However, the roughness of the operand sets can not uniquely bound the roughness of their intersection set. The results presented here show that a bound of the set operation can be determined from their operand's roughnesses under some operations. We provide an example to show the derived bounds from operand's roughness.

To date, because of the computational complexity of using a general type-2 fuzzy set (T2 FS) in a T2 fuzzy logic system (FLS), most people only use an interval T2 FS, the result being an interval T2 FLS (IT2 FLS). Unfortunately, there is a heavy educational burden even to using an IT2 FLS. This burden has to do with first having to learn general T2 FS mathematics, and then specializing it to an IT2 FSs. In retrospect, we believe that requiring a person to use T2 FS mathematics represents a barrier to the use of an IT2 FLS. In this paper, we demonstrate that it is unnecessary to take the route from general T2 FS to IT2 FS, and that all of the results that are needed to implement an IT2 FLS can be obtained using T1 FS mathematics. As such, this paper is a novel tutorial that makes an IT2 FLS much more accessible to all readers of this journal. We can now develop an IT2 FLS in a much more straightforward way

This work introduces representations and methods that improve the inferencing speeds of type-2 fuzzy logic systems by exploiting methods from computational geometry. A new representation of a secondary membership function is given along with its operations. We give the results from some initial experiments looking at the inferencing speed of a small, limited system. The results show a four and a half fold increase in the inferencing speed of our methods over a comparable standard system. Our motivation is that by improving the performance of type-2 systems such technologies may become more widely applied

This paper describes a fuzzy approach to computer-aided medical diagnosis in a clinical context. It introduces a formal view of diagnosis in clinical settings and shows the relevance and possible uses of fuzzy cognitive maps. A constraint satisfaction method is introduced that uses the temporal uncertainty in symptom durations that may occur with particular diseases. The method results in an estimate of the stage of the disease if the temporal constraints of the disease in relation to the occurrence of the symptoms are satisfied. A lightweight fuzzy process is described and evaluated in the context of diagnosis of two confusable diseases. The process is based on the idea of an incremental simple additive model for fuzzy sets supporting and negating particular diseases. These are combined to produce an index of support for a particular disease. The process is developed to allow fuzzy symptom information on the intensity and duration of symptoms. Results are presented showing the effectiveness of the method for supporting differential diagnosis.

Type-2 fuzzy logic systems require considerable processing resources to arrive at a conclusion. This paper shows how this can be significantly reduced, focusing specifically on the meet operation. We show how to eliminate large amounts of redundancy from the operation and then present a geometric method that is even more efficient. Results from initial experiments into performance increases are given. These results show that our method can give a significant increase in performance for certain types of secondary memberships.

Terms used in fuzzy systems are almost invariably normalised, convex and distinct. The shapes of these terms are generated by certain accepted membership functions: piecewise linear functions, Gaussians or Sigmoids are almost exclusively used. This paper extends previous work in which it was suggested that non-convex membership functions might be considered for use in the context of modelling human decision making utilising fuzzy expert systems. In particular, the merits of non-convex fuzzy sets are discussed and a case study is presented to investigate inferencing with non-convex fuzzy sets in a practical implementation. It is shown that it is indeed possible to build a fuzzy expert system featuring usual Mamdani style fuzzy inference in which a time-related non-convex fuzzy set is used together with 'traditional' fuzzy sets. An examination is made of the resultant output surface generated by four different sub-classes of non-convex membership functions.

The shapes of terms used in fuzzy systems have adopted several 'conventions'. Terms are almost invariably normalised (having a maximum membership value of 1), convex (having a single maximum or plateau maxima) and distinct (being restricted in their degree of overlap: often expressed as some variation on the concept that all membership values at any point in the universe of discourse sum to I across that universe). The shape of these terms are generated by certain accepted membership functions: piecewise linear functions (with restrictions), Gaussians or Sigmoids are almost exclusively used. As such these constitute only a small subset of the total set of possible shapes of terms. These conventions are largely empirical or are justified by arguments based on what might loosely be called 'fuzzy control principles'. The paper highlights a number of membership functions that developers of fuzzy systems outside the paradigm of fuzzy control may consider as alternatives. In particular, we highlight subsumed fuzzy sets, discuss the merits of non-convex fuzzy sets and present a medical application where sub-normal fuzzy sets have been used. These ideas are reinforced by examples.

This paper describes a computationally fast and simple algorithm that produces an index of support for the diagnosis of diseases based on type-2 fuzzy sets. The procedure allows the specification of fuzzy symptoms by using fuzzy relations to determine a symptom's strength based on its observed intensity and the certainty of its duration. We formally describe the fuzzy approach and the algorithm, and show how it relates to a disease given a range of symptom strengths.

Type-2 fuzzy sets are growing in popularity as, for certain applications they model uncertainty and imprecision better than type-1 fuzzy sets. However, type-2 fuzzy sets can be difficult to understand and explain. Recent work has introduced embedded type-2 fuzzy sets and the representation theorem which enable us to discuss type-2 fuzzy sets in a different way. In particular they allow for alternative proofs of theoretical results which are often easier to follow. Interval valued type-2 fuzzy sets are found to be useful in many application. This paper presents embedded interval valued type-2 fuzzy sets to aid the discussion of interval valued type-2 fuzzy sets and to prove the join and meet of interval valued type-2 fuzzy sets.

Type-2 fuzzy sets let us model and minimize the effects of uncertainties in rule-base fuzzy logic systems. However, they are difficult to understand for a variety of reasons which we enunciate. In this paper, we strive to overcome the difficulties by: (1) establishing a small set of terms that let us easily communicate about type-2 fuzzy sets and also let us define such sets very precisely, (2) presenting a new representation for type-2 fuzzy sets, and (3) using this new representation to derive formulas for union, intersection and complement of type-2 fuzzy sets without having to use the Extension Principle.

The paper reports on the use of modular neural networks for predicting corporate bankruptcy. We obtained our financial, as well as, political and economic data from The London Stock Exchange, JORDANS financial database of major British public and private companies, and the Bank of England. In the past, various statistical techniques, such as univariate and multivariate discriminant analysis have been used in the modelling of corporate bankruptcy prediction. We use domain expert knowledge to select, and organise data in the modular neural network architecture constructed for this study. There are three sub-networks representing the periods, 1994, 1995, and 1996. Each sub-network is made of five adjacent networks representing the Balance Sheet network, the Profit and Loss network, the Financial Summary network, the Key Financial Ratios network, and the Economic and Political factors network. These adjacent networks although coupled but not linked at the input level represent five facets of failure in predicting corporate bankruptcy. The training sets represent data for 2500 companies selected randomly from a population of 270000 sample. The trained neural network will access 435000 data records before making a prediction for the particular company. The results obtained shows that neural networks outperform statistical techniques in modelling corporate failure prediction.

Type 2 fuzzy sets allow for linguistic grades of membership and, therefore, present a better representation of the 'fuzziness', when applied to a particular problem, than type 1 fuzzy sets. However, the associated cost is that the fuzzy membership grades and rules have somehow to be determined and no recognised approach yet exists. For type 1 systems a number of approaches have been adopted. One in particular is the adaptive network based fuzzy inferencing system (ANFIS) which has successfully been applied to a variety of applications. ANFIS takes domain data and learns the membership functions and rules for a type 1 fuzzy inferencing system. Our work aims to extend this approach for type 2 systems. Our Type 2 adaptive fuzzy inferencing system has inputs that are linguistic variables and the membership functions for these fuzzy grades are learnt from the relationship between these inputs and the given output. The paper describes the algorithm developed highlighting the theoretical and computational issues involved.

This paper provides a guide and tutorial to type 2 fuzzy sets. Type 2 fuzzy sets allow for linguistic grades of membership thus assisting in knowledge representation. They also offer improvement on inferencing with type 1 sets. The various approaches to knowledge representation and inferencing are discussed, with worked examples, and some of the applications of type 2 sets are reported.