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One of the most preeminent ways of applying mathematics in real-world scenario modeling involves graph theory. A graph can be undirected or directed depending on whether the pairwise relationships among objects are symmetric or not. Nevertheless, in many real-world situations, representing a set of complex relational objects as directed or undirected is not su¢ cient. Weighted graphs o§er a framework that helps to over come certain conceptual limitations. We show using the concept of an isomorphism that weighted graphs have a natural connection to fuzzy graphs. As we show in the book, this allows results to be carried back and forth between weighted graphs and fuzzy graphs. This idea is in keeping with the important paper by Klement and Mesiar that shows that many families of fuzzy sets are lattice isomorphic to each other. We also outline the important work of Head and Weinberger that show how results from ordinary mathematics can be carried over to fuzzy mathematics. We focus on the concepts connectivity, degree sequences and saturation, and intervals and gates in weighted graphs.

As a new interpretable model, three-way decision has also received academic attention in machine learning. With respect to different hesitant fuzzy information, this book deeply discusses the deduction process of decision rules of three-way decision and generates interpretable knowledge with the risk semantics. It further explores the applications of three-way decision to support healthcare management. This book is used as a reference for engineers, technicians, and researchers who are working in the fields of management science, operation management, computer science, information management, fuzzy mathematics, business intelligence, and other fields. It also serves as a textbook for postgraduate and senior undergraduate students of the relevant professional institutions of higher learning.

This book presents an extension of fuzzy set theory allowing for multi-polar information, discussing its impact on the theoretical and practical development of multi-criteria decision making. It reports on set of hybrid models developed by the authors, and show how they can be adapted, case by case, to the lack of certainty under a variety of criteria. Among them, hybrid models combining m-polar fuzzy sets with rough, soft and 2-tuple linguistic sets, and m-polar hesitant fuzzy sets and hesitant m-polar fuzzy are presented, together with some significant applications. In turn, outranking decision-making techniques such as m-polar fuzzy ELECTRE I, II, III and IV methods, as well as m-polar fuzzy PROMETHEE I and II methods, are developed. The efficiency of these decision-making procedures, as well as other possible extensions studied by the authors, is shown in some real-world applications. Overall, this book offers a guide on methodologies to deal with the multi-polarity and fuzziness of the real-world problems, simultaneously. By including algorithms and computer programming codes, it provides a practice-oriented reference guide to both researchers and professionals working at the interface between computational intelligence and decision making.

This book is considered as a monograph but also as a potential textbook for graduate students, focusing on the application of FCMs for modelling and analysing the behaviour of multicomponent systems. In the last two decades, no monograph or textbook has been published on the topic Fuzzy Cognitive Maps (FCM), so this new book is definitely filling a gap in the literature of computational intelligence. The book is built up didactically, the novel results in the field being presented in the way of starting with two real-life case studies, one in the area of waste management, while the other one in modelling bank management systems. In both cases, the book starts with explaining the applied problem and then presenting how the model construction is done and what problems emerge when attempts are made for applying directly earlier results on FCM modelling. In the first case study, the problem of the oversimplification leads to inadequacy of the model, and then it is shown how new,much finer models can be built up based on expert domain knowledge. Then, the new problem of losing transparency and interpretability emerges, and as a solution, a new algorithm family is proposed that reduces FCMs to fewer components, while preserving the essential characteristics of the original model.The second case study raises the problems of stability and sensitivity of FCMs, especially, considering that expert knowledge is often uncertain and subjective. The new results summarised in the book target the questions of how to ascertain whether an FCM is converging to one or several fixed point attractors, whether there is a bifurcation when parameters are changing, etc. Both problems deal with the ultimate question whether the system modelled is stable and sustainable.