Interval type-2 Atanassov-intuitionistic fuzzy logic for uncertainty modelling

This thesis investigates a new paradigm for uncertainty modelling by employing a new class of type-2 fuzzy logic system that utilises fuzzy sets with membership and non-membership functions that are intervals. Fuzzy logic systems, employing type-1 fuzzy sets, that mark a shift from computing with numbers towards computing with words have made remarkable impacts in the field of artificial intelligence. Fuzzy logic systems of type-2, a generalisation of type-1 fuzzy logic systems that utilise type-2 fuzzy sets, have created tremendous advances in uncertainty modelling. The key feature of the type-2 fuzzy logic systems, with particular reference to interval type-2 fuzzy logic systems, is that the membership functions of interval type-2 fuzzy sets are themselves fuzzy. These give interval type-2 fuzzy logic systems an advantage over their type-1 counterparts which have precise membership functions. Whilst the interval type-2 fuzzy logic systems are effective in modelling uncertainty, they are not able to adequately handle an indeterminate/neutral characteristic of a set, because interval type-2 fuzzy sets are only specified by membership functions with an implicit assertion that the non-membership functions are complements of the membership functions (lower or upper). In a real life scenario, it is not necessarily the case that the non-membership function of a set is complementary to the membership function. There may be some degree of hesitation arising from ignorance or a complete lack of interest concerning a particular phenomenon. Atanassov intuitionistic fuzzy set, another generalisation of the classical fuzzy set, captures this thought process by simultaneously defining a fuzzy set with membership and non-membership functions such that the sum of both membership and non-membership functions is less than or equal to 1. In this thesis, the advantages of both worlds (interval type-2 fuzzy set and Atanassov intuitionistic fuzzy set) are explored and a new and enhanced class of interval type-2 fuzzy set namely, interval type-2 Atanassov intuitionistic fuzzy set, that enables hesitation, is introduced. The corresponding fuzzy logic system namely, interval type-2 Atanassov intuitionistic fuzzy logic system is rigorously and systematically formulated. In order to assess this thesis investigates a new paradigm for uncertainty modelling by employing a new class of type-2 fuzzy logic system that utilises fuzzy sets with membership and non-membership functions that are intervals. Fuzzy logic systems, employing type-1 fuzzy sets, that mark shift from computing with numbers towards computing with words have made remarkable impacts in the field of artificial intelligence. Fuzzy logic systems of type-2, a generalisation of type-1 fuzzy logic systems that utilise type-2 fuzzy sets, have created tremendous advances in uncertainty modelling. The key feature of the type-2 fuzzy logic systems, with particular reference to interval type-2 fuzzy logic systems, is that the membership functions of interval type-2 fuzzy sets are themselves fuzzy. These give interval type-2 fuzzy logic systems an advantage over their type-1 counterparts which have precise membership functions. Whilst the interval type-2 fuzzy logic systems are effective in modelling uncertainty, they are not able to adequately handle an indeterminate/neutral characteristic of a set, because interval type-2 fuzzy sets are only specified by membership functions with an implicit assertion that the non-membership functions are complements of the membership functions (lower or upper). In a real life scenario, it is not necessarily the case that the non-membership function of a set is complementary to the membership function. There may be some degree of hesitation arising from ignorance or a complete lack of interest concerning a particular phenomenon. Atanassov intuitionistic fuzzy set, another generalisation of the classical fuzzy set, captures this thought process by simultaneously defining a fuzzy set with membership and non-membership functions such that the sum of both membership and non-membership functions is less than or equal to 1. In this thesis, the advantages of both worlds (interval type-2 fuzzy set and Atanassov intuitionistic fuzzy set) are explored and a new and enhanced class of interval type-2 fuzz set namely, interval type-2 Atanassov intuitionistic fuzzy set, that enables hesitation, is introduced. The corresponding fuzzy logic system namely, interval type-2 Atanassov intuitionistic fuzzy logic system is rigorously and systematically formulated. In order to assess the viability and efficacy of the developed framework, the possibilities of the optimisation of the parameters of this class of fuzzy systems are rigorously examined. First, the parameters of the developed model are optimised using one of the most popular fuzzy logic optimisation algorithms such as gradient descent (first-order derivative) algorithm and evaluated on publicly available benchmark datasets from diverse domains and characteristics. It is shown that the new interval type-2 Atanassov intuitionistic fuzzy logic system is able to handle uncertainty well through the minimisation of the error of the system compared with other approaches on the same problem instances and performance criteria. Secondly, the parameters of the proposed framework are optimised using a decoupledextended Kalman filter (second-order derivative) algorithm in order to address the shortcomings of the first-order gradient descent method. It is shown statistically that the performance of this new framework with fuzzy membership and non-membership functions is significantly better than the classical interval type-2 fuzzy logic systems which have only the fuzzy membership functions, and its type-1 counterpart which are specified by single membership and non-membership functions. The model is also assessed using a hybrid learning of decoupled extended Kalman filter and gradient descent methods. The proposed framework with hybrid learning algorithm is evaluated by comparing it with existing approaches reported in the literature on the same problem instances and performance metrics. The simulation results have demonstrated the potential benefits of using the proposed framework in uncertainty modelling. In the overall, the fusion of these two concepts (interval type-2 fuzzy logic system and Atanassov intuitionistic fuzzy logic system) provides a synergistic capability in dealing with imprecise and vague information

A review of application of multi-criteria decision making methods in construction

Construction is an area of study wherein making decisions adequately can mean the difference between success and failure. Moreover, most of the activities belonging to this sector involve taking into account a large number of conflicting aspects, which hinders their management as a whole. Multi-criteria decision making analysis arose to model complex problems like these. This paper reviews the application of 22 different methods belonging to this discipline in various areas of the construction industry clustered in 11 categories. The most significant methods are briefly discussed, pointing out their principal strengths and limitations. Furthermore, the data gathered while performing the paper are statistically analysed to identify different trends concerning the use of these techniques. The review shows their usefulness in characterizing very different decision making environments, highlighting the reliability acquired by the most pragmatic and widespread methods and the emergent tendency to use some of them in combination

Interval type-2 Atanassov-intuitionistic fuzzy logic for uncertainty modelling

This thesis investigates a new paradigm for uncertainty modelling by employing a new class of type-2 fuzzy logic system that utilises fuzzy sets with membership and non-membership functions that are intervals. Fuzzy logic systems, employing type-1 fuzzy sets, that mark a shift from computing with numbers towards computing with words have made remarkable impacts in the field of artificial intelligence. Fuzzy logic systems of type-2, a generalisation of type-1 fuzzy logic systems that utilise type-2 fuzzy sets, have created tremendous advances in uncertainty modelling. The key feature of the type-2 fuzzy logic systems, with particular reference to interval type-2 fuzzy logic systems, is that the membership functions of interval type-2 fuzzy sets are themselves fuzzy. These give interval type-2 fuzzy logic systems an advantage over their type-1 counterparts which have precise membership functions. Whilst the interval type-2 fuzzy logic systems are effective in modelling uncertainty, they are not able to adequately handle an indeterminate/neutral characteristic of a set, because interval type-2 fuzzy sets are only specified by membership functions with an implicit assertion that the non-membership functions are complements of the membership functions (lower or upper). In a real life scenario, it is not necessarily the case that the non-membership function of a set is complementary to the membership function. There may be some degree of hesitation arising from ignorance or a complete lack of interest concerning a particular phenomenon. Atanassov intuitionistic fuzzy set, another generalisation of the classical fuzzy set, captures this thought process by simultaneously defining a fuzzy set with membership and non-membership functions such that the sum of both membership and non-membership functions is less than or equal to 1. In this thesis, the advantages of both worlds (interval type-2 fuzzy set and Atanassov intuitionistic fuzzy set) are explored and a new and enhanced class of interval type-2 fuzzy set namely, interval type-2 Atanassov intuitionistic fuzzy set, that enables hesitation, is introduced. The corresponding fuzzy logic system namely, interval type-2 Atanassov intuitionistic fuzzy logic system is rigorously and systematically formulated. In order to assess this thesis investigates a new paradigm for uncertainty modelling by employing a new class of type-2 fuzzy logic system that utilises fuzzy sets with membership and non-membership functions that are intervals. Fuzzy logic systems, employing type-1 fuzzy sets, that mark shift from computing with numbers towards computing with words have made remarkable impacts in the field of artificial intelligence. Fuzzy logic systems of type-2, a generalisation of type-1 fuzzy logic systems that utilise type-2 fuzzy sets, have created tremendous advances in uncertainty modelling. The key feature of the type-2 fuzzy logic systems, with particular reference to interval type-2 fuzzy logic systems, is that the membership functions of interval type-2 fuzzy sets are themselves fuzzy. These give interval type-2 fuzzy logic systems an advantage over their type-1 counterparts which have precise membership functions. Whilst the interval type-2 fuzzy logic systems are effective in modelling uncertainty, they are not able to adequately handle an indeterminate/neutral characteristic of a set, because interval type-2 fuzzy sets are only specified by membership functions with an implicit assertion that the non-membership functions are complements of the membership functions (lower or upper). In a real life scenario, it is not necessarily the case that the non-membership function of a set is complementary to the membership function. There may be some degree of hesitation arising from ignorance or a complete lack of interest concerning a particular phenomenon. Atanassov intuitionistic fuzzy set, another generalisation of the classical fuzzy set, captures this thought process by simultaneously defining a fuzzy set with membership and non-membership functions such that the sum of both membership and non-membership functions is less than or equal to 1. In this thesis, the advantages of both worlds (interval type-2 fuzzy set and Atanassov intuitionistic fuzzy set) are explored and a new and enhanced class of interval type-2 fuzz set namely, interval type-2 Atanassov intuitionistic fuzzy set, that enables hesitation, is introduced. The corresponding fuzzy logic system namely, interval type-2 Atanassov intuitionistic fuzzy logic system is rigorously and systematically formulated. In order to assess the viability and efficacy of the developed framework, the possibilities of the optimisation of the parameters of this class of fuzzy systems are rigorously examined. First, the parameters of the developed model are optimised using one of the most popular fuzzy logic optimisation algorithms such as gradient descent (first-order derivative) algorithm and evaluated on publicly available benchmark datasets from diverse domains and characteristics. It is shown that the new interval type-2 Atanassov intuitionistic fuzzy logic system is able to handle uncertainty well through the minimisation of the error of the system compared with other approaches on the same problem instances and performance criteria. Secondly, the parameters of the proposed framework are optimised using a decoupledextended Kalman filter (second-order derivative) algorithm in order to address the shortcomings of the first-order gradient descent method. It is shown statistically that the performance of this new framework with fuzzy membership and non-membership functions is significantly better than the classical interval type-2 fuzzy logic systems which have only the fuzzy membership functions, and its type-1 counterpart which are specified by single membership and non-membership functions. The model is also assessed using a hybrid learning of decoupled extended Kalman filter and gradient descent methods. The proposed framework with hybrid learning algorithm is evaluated by comparing it with existing approaches reported in the literature on the same problem instances and performance metrics. The simulation results have demonstrated the potential benefits of using the proposed framework in uncertainty modelling. In the overall, the fusion of these two concepts (interval type-2 fuzzy logic system and Atanassov intuitionistic fuzzy logic system) provides a synergistic capability in dealing with imprecise and vague information

Using Pythagorean Fuzzy Sets (PFS) in Multiple Criteria Group Decision Making (MCGDM) Methods for Engineering Materials Selection Applications

The process of materials’ selection is very critical during the initial stages of designing manufactured products. Inefficient decision-making outcomes in the material selection process could result in poor quality of products and unnecessary costs. In the last century, numerous materials have been developed for manufacturing mechanical components in different industries. Many of these new materials are similar in their properties and performances, thus creating great challenges for designers and engineers to make accurate selections. Our main objective in this work is to assist decision makers (DMs) within the manufacturing field to evaluate materials alternatives and to select the best alternative for specific manufacturing purposes. In this research, new hybrid fuzzy Multiple Criteria Group Decision Making (MCGDM) methods are proposed for the material selection problem. The proposed methods tackle some challenges that are associated with the material selection decision making process, such as aggregating decision makers’ (DMs) decisions appropriately and modeling uncertainty. In the proposed hybrid models, a novel aggregation approach is developed to convert DMs crisp decisions to Pythagorean fuzzy sets (PFS). This approach gives more flexibility to DMs to express their opinions than the traditional fuzzy and intuitionistic sets (IFS). Then, the proposed aggregation approach is integrated with a ranking method to solve the Pythagorean Fuzzy Multi Criteria Decision Making (PFMCGDM) problem and rank the material alternatives. The ranking methods used in the hybrid models are the Pythagorean Fuzzy TOPSIS (The Technique for Order of Preference by Similarity to Ideal Solution) and Pythagorean Fuzzy COPRAS (COmplex PRoportional Assessment). TOPSIS and COPRAS are selected based on their effectiveness and practicality in dealing with the nature of material selection problems. In the aggregation approach, the Sugeno Fuzzy measure and the Shapley value are used to fairly distribute the DMs weight in the Pythagorean Fuzzy numbers. Additionally, new functions to calculate uncertainty from DMs recommendations are developed using the Takagai-Sugeno approach. The literature reveals some work on these methods, but to our knowledge, there are no published works that integrate the proposed aggregation approach with the selected MCDM ranking methods under the Pythagorean Fuzzy environment for the use in materials selection problems. Furthermore, the proposed methods might be applied, due to its novelty, to any MCDM problem in other areas. A practical validation of the proposed hybrid PFMCGDM methods is investigated through conducting a case study of material selection for high pressure turbine blades in jet engines. The main objectives of the case study were: 1) to investigate the new developed aggregation approach in converting real DMs crisp decisions into Pythagorean fuzzy numbers; 2) to test the applicability of both the hybrid PFMCGDM TOPSIS and the hybrid PFMCGDM COPRAS methods in the field of material selection. In this case study, a group of five DMs, faculty members and graduate students, from the Materials Science and Engineering Department at the University of Wisconsin-Milwaukee, were selected to participate as DMs. Their evaluations fulfilled the first objective of the case study. A computer application for material selection was developed to assist designers and engineers in real life problems. A comparative analysis was performed to compare the results of both hybrid MCGDM methods. A sensitivity analysis was conducted to show the robustness and reliability of the outcomes obtained from both methods. It is concluded that using the proposed hybrid PFMCGDM TOPSIS method is more effective and practical in the material selection process than the proposed hybrid PFMCGDM COPRAS method. Additionally, recommendations for further research are suggested

Fuzzy Sets, Fuzzy Logic and Their Applications

The present book contains 20 articles collected from amongst the 53 total submitted manuscripts for the Special Issue “Fuzzy Sets, Fuzzy Loigic and Their Applications” of the MDPI journal Mathematics. The articles, which appear in the book in the series in which they were accepted, published in Volumes 7 (2019) and 8 (2020) of the journal, cover a wide range of topics connected to the theory and applications of fuzzy systems and their extensions and generalizations. This range includes, among others, management of the uncertainty in a fuzzy environment; fuzzy assessment methods of human-machine performance; fuzzy graphs; fuzzy topological and convergence spaces; bipolar fuzzy relations; type-2 fuzzy; and intuitionistic, interval-valued, complex, picture, and Pythagorean fuzzy sets, soft sets and algebras, etc. The applications presented are oriented to finance, fuzzy analytic hierarchy, green supply chain industries, smart health practice, and hotel selection. This wide range of topics makes the book interesting for all those working in the wider area of Fuzzy sets and systems and of fuzzy logic and for those who have the proper mathematical background who wish to become familiar with recent advances in fuzzy mathematics, which has entered to almost all sectors of human life and activity

Fuzzy Systems

This book presents some recent specialized works of theoretical study in the domain of fuzzy systems. Over eight sections and fifteen chapters, the volume addresses fuzzy systems concepts and promotes them in practical applications in the following thematic areas: fuzzy mathematics, decision making, clustering, adaptive neural fuzzy inference systems, control systems, process monitoring, green infrastructure, and medicine. The studies published in the book develop new theoretical concepts that improve the properties and performances of fuzzy systems. This book is a useful resource for specialists, engineers, professors, and students

Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets

Neutrosophy (1995) is a new branch of philosophy that studies triads of the form (, , ), where is an entity {i.e. element, concept, idea, theory, logical proposition, etc.}, is the opposite of , while is the neutral (or indeterminate) between them, i.e., neither nor .Based on neutrosophy, the neutrosophic triplets were founded, which have a similar form (x, neut(x), anti(x)), that satisfy several axioms, for each element x in a given set.This collective book presents original research papers by many neutrosophic researchers from around the world, that report on the state-of-the-art and recent advancements of neutrosophic triplets, neutrosophic duplets, neutrosophic multisets and their algebraic structures – that have been defined recently in 2016 but have gained interest from world researchers. Connections between classical algebraic structures and neutrosophic triplet / duplet / multiset structures are also studied. And numerous neutrosophic applications in various fields, such as: multi-criteria decision making, image segmentation, medical diagnosis, fault diagnosis, clustering data, neutrosophic probability, human resource management, strategic planning, forecasting model, multi-granulation, supplier selection problems, typhoon disaster evaluation, skin lesson detection, mining algorithm for big data analysis, etc