Constrained interval type-2 fuzzy sets

In many contexts, type-2 fuzzy sets 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 that can be considered acceptable. This leads, usually, to the loss of the semantic relationship between the type-2 fuzzy set and the concept it models. As a consequence, the interpretability of some of the embedded sets and the explainability of the uncertainty measures obtained from them can decrease. To overcome these issues, constrained type-2 fuzzy sets have been proposed. However, no formal definitions for some of their key components (e.g. acceptable embedded sets) and constrained operations have been given. The goal of this paper is to provide some theoretical underpinning for the definition of constrained type-2 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

Novel techniques for modelling uncertain human reasoning in explainable artificial intelligence

In recent years, there has been a growing need for intelligent systems that not only are able to provide reliable predictions but can also produce explanations for their outputs. 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 design models with a transparent decision process, from which it is possible to extract human-understandable explanations. The use of interval type-2 fuzzy logic in the XAI field, however, is limited: the improved performances of interval type-2 fuzzy systems and their ability to handle a higher degree of uncertainty comes at the cost of increased complexity that makes the semantic mapping between the input and outputs harder to understand intuitively. The presence of type-reduction, in some contexts fail to preserve the semantic value of the fuzzy sets and rules involved in the decision process. By semantic value, we specifically refer to the capacity of interpreting the output of the fuzzy system in respect to the pre-defined and thus understood linguistic variables used for the antecedents and consequents of the system. An attempt at increasing the explainability of interval type-2 fuzzy logic was first established by Garibaldi and Guadarrama in 2011, with the introduction of constrained type-2 fuzzy sets. However, extensive work needs to be carried out to develop the algorithms necessary for their practical use in fuzzy systems. The aim of this thesis is to extend the initial work on constrained interval type-2 fuzzy sets to develop a framework that preserves the semantic value throughout the modelling and decision process. Achieving this goal would allow the creation of a new class of fuzzy systems that show additional interpretable properties, and could further encourage the use of interval type-2 fuzzy logic in XAI. After the formal definition of the required components and theorems, different approaches are explored to develop inference algorithms that preserve the semantic value of the sets during the input-output mapping, while keeping reasonable run-times on modern computer hardware. The novel frameworks are then tested in a series of practical applications from the real world, in order to assess both their prediction performances and show the quality of the explanations these models can generate. Finally, the original definitions of constrained intervals type-2 fuzzy sets are refined to produce a novel approach which combines uncertain data and represents them using intuitive constrained interval type-2 fuzzy sets. Overall, as a result of the work presented here, it is now possible to design constrained interval type-2 fuzzy systems that preserve the enhanced semantic value provided by constrained interval-type-2 fuzzy sets throughout the inference, type-reduction and defuzzification stages. This characteristic is then used to improve the semantic interpretability of the system outputs, making constrained interval type-2 fuzzy systems a valuable alternative to interval type-2 fuzzy systems in XAI. The research presented here has resulted in three journal articles, two of which have already been published in IEEE Transactions on Fuzzy Systems, and four papers presented at the FUZZ-IEEE international conference between 2018 and 2020

Modeling and Evaluation of Single Machine Flexibility Using Fuzzy Entropy and Genetic Algorithm Based Approach

International audienceFlexibility has long been recognized as a manufacturing capability that has the potential to impact mainly the competitive position of an organization. The entropy approach, which was extended from information theory, fell in handling problems with incomplete and uncertain data, because it depicts only the stochastic aspects included with measured observations. In order to get a global view, this work proposes a new approach based on fuzzy entropy concept. The development of the fuzzy model results in a set of nonlinear constrained problems to be solved using a metaheuristics method. The applicability of our approach is illustrated through a flexible manufacturing cell. By adopting such framework, both dimensions of uncertainty in system modeling, expressed by stochastic variability and imprecision, can be taken into consideration

Novel techniques for modelling uncertain human reasoning in explainable artificial intelligence

In recent years, there has been a growing need for intelligent systems that not only are able to provide reliable predictions but can also produce explanations for their outputs. 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 design models with a transparent decision process, from which it is possible to extract human-understandable explanations. The use of interval type-2 fuzzy logic in the XAI field, however, is limited: the improved performances of interval type-2 fuzzy systems and their ability to handle a higher degree of uncertainty comes at the cost of increased complexity that makes the semantic mapping between the input and outputs harder to understand intuitively. The presence of type-reduction, in some contexts fail to preserve the semantic value of the fuzzy sets and rules involved in the decision process. By semantic value, we specifically refer to the capacity of interpreting the output of the fuzzy system in respect to the pre-defined and thus understood linguistic variables used for the antecedents and consequents of the system. An attempt at increasing the explainability of interval type-2 fuzzy logic was first established by Garibaldi and Guadarrama in 2011, with the introduction of constrained type-2 fuzzy sets. However, extensive work needs to be carried out to develop the algorithms necessary for their practical use in fuzzy systems. The aim of this thesis is to extend the initial work on constrained interval type-2 fuzzy sets to develop a framework that preserves the semantic value throughout the modelling and decision process. Achieving this goal would allow the creation of a new class of fuzzy systems that show additional interpretable properties, and could further encourage the use of interval type-2 fuzzy logic in XAI. After the formal definition of the required components and theorems, different approaches are explored to develop inference algorithms that preserve the semantic value of the sets during the input-output mapping, while keeping reasonable run-times on modern computer hardware. The novel frameworks are then tested in a series of practical applications from the real world, in order to assess both their prediction performances and show the quality of the explanations these models can generate. Finally, the original definitions of constrained intervals type-2 fuzzy sets are refined to produce a novel approach which combines uncertain data and represents them using intuitive constrained interval type-2 fuzzy sets. Overall, as a result of the work presented here, it is now possible to design constrained interval type-2 fuzzy systems that preserve the enhanced semantic value provided by constrained interval-type-2 fuzzy sets throughout the inference, type-reduction and defuzzification stages. This characteristic is then used to improve the semantic interpretability of the system outputs, making constrained interval type-2 fuzzy systems a valuable alternative to interval type-2 fuzzy systems in XAI. The research presented here has resulted in three journal articles, two of which have already been published in IEEE Transactions on Fuzzy Systems, and four papers presented at the FUZZ-IEEE international conference between 2018 and 2020

Combining clusterings in the belief function framework

International audienceIn this paper, we propose a clustering ensemble method based on Dempster-Shafer Theory. In the first step, base partitions are generated by evidential clustering algorithms such as the evidential c-means or EVCLUS. Base credal partitions are then converted to their relational representations, which are combined by averaging. The combined relational representation is then made transitive using the theory of intuitionistic fuzzy relations. Finally, the consensus solution is obtained by minimizing an error function. Experiments with simulated and real datasets show the good performances of this method

Interval and fuzzy optimization. Applications to data envelopment analysis

Enhancing concern in the efficiency assessment of a set of peer entities termed Decision Making Units (DMUs) in many fields from industry to healthcare has led to the development of efficiency assessment models and tools. Data Envelopment Analysis (DEA) is one of the most important methodologies to measure efficiency assessment through the comparison of a group of DMUs. It permits the use of multiple inputs/outputs without any functional form. It is vastly applied to production theory in Economics and benchmarking in Operations Research. In conventional DEA models, the observed inputs and outputs possess precise and realvalued data. However, in the real world, some problems consider imprecise and integer data. For example, the number of defect-free lamps, the fleet size, the number of hospital beds or the number of staff can be represented in some cases as imprecise and integer data. This thesis considers several novel approaches for measuring the efficiency assessment of DMUs where the inputs and outputs are interval and fuzzy data. First, an axiomatic derivation of the fuzzy production possibility set is presented and a fuzzy enhanced Russell graph measure is formulated using a fuzzy arithmetic approach. The proposed approach uses polygonal fuzzy sets and LU-fuzzy partial orders and provides crisp efficiency measures (and associated efficiency ranking) as well as fuzzy efficient targets. The second approach is a new integer interval DEA, with the extension of the corresponding arithmetic and LU-partial orders to integer intervals. Also, a new fuzzy integer DEA approach for efficiency assessment is presented. The proposed approach considers a hybrid scenario involving trapezoidal fuzzy integer numbers and trapezoidal fuzzy numbers. Fuzzy integer arithmetic and partial orders are introduced. Then, using appropriate axioms, a fuzzy integer DEA technology can be derived. Finally, an inverse DEA based on the non-radial slacks-based model in the presence of uncertainty, employing both integer and continuous interval data is presented

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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