Study on Rough Sets and Fuzzy Sets in Constructing Intelligent Information System

Since human being is not an omniscient and omnipotent being, we are actually living in an uncertain world. Uncertainty was involved and connected to every aspect of human life as a quotation from Albert Einstein said: �As far as the laws of mathematics refer to reality, they are not certain. And as far as they are certain, they do not refer to reality.� The most fundamental aspect of this connection is obviously shown in human communication. Naturally, human communication is built on the perception1-based information instead of measurement-based information in which perceptions play a central role in human cognition [Zadeh, 2000]. For example, it is naturally said in our communication that �My house is far from here.� rather than let say �My house is 12,355 m from here�. Perception-based information is a generalization of measurement-based information, where perception-based information such as �John is excellent.� is hard to represent by measurement-based version. Perceptions express human subjective view. Consequently, they tend to lead up to misunderstanding. Measurements then are needed such as defining units of length, time, etc., to provide objectivity as a means to overcome misunderstanding. Many measurers were invented along with their methods and theories of measurement. Hence, human cannot communicate with measurers including computer as a product of measurement era unless he uses measurement-based information. Perceptions are intrinsic aspect in uncertainty-based information. In this case, information may be incomplete, imprecise, fragmentary, not fully reliable, vague, contradictory, or deficient in some other way. 1In psychology, perception is understood as a process of translating sensory stimulation into an organized experience Generally, these various information deficiencies may express different types of uncertainty. It is necessary to construct a computer-based information system called intelligent information system that can process uncertainty-based information. In the future, computers are expected to be able to make communication with human in the level of perception. Many theories were proposed to express and process the types of uncertainty such as probability, possibility, fuzzy sets, rough sets, chaos theory and so on. This book extends and generalizes existing theory of rough set, fuzzy sets and granular computing for the purpose of constructing intelligent information system. The structure of this book is the following: In Chapter 2, types of uncertainty in the relation to fuzziness, probability and evidence theory (belief and plausibility measures) are briefly discussed. Rough set regarded as another generalization of crisp set is considered to represent rough event in the connection to the probability theory. Special attention will be given to formulation of fuzzy conditional probability relation generated by property of conditional probability of fuzzy event. Fuzzy conditional probability relation then is used to represent similarity degree of two fuzzy labels. Generalization of rough set induced by fuzzy conditional probability relation in terms of covering of the universe is given in Chapter 3. In the relation to fuzzy conditional probability relation, it is necessary to consider an interesting mathematical relation called weak fuzzy similarity relation as a generalization of fuzzy similarity relation proposed by Zadeh [1995]. Fuzzy rough set and generalized fuzzy rough set are proposed along with the generalization of rough membership function. Their properties are examined. Some applications of these methods in information system such as α-redundancy of object and dependency of domain attributes are discussed. In addition, multi rough sets based on multi-context of attributes in the presence of multi-contexts information system is defined and proposed in Chapter 4. In the real application, depending on the context, a given object may have different values of attributes. In other words, set of attributes might be represented based on different context, where they may provide different values for a given object. Context can be viewed as background or situation in which somehow it is necessary to group some attributes as a subset of attributes and consider the subset as a context. Finally, Chapter 5 summarizes all discussed in this book and puts forward some future topics of research

Isomorphic multiplicative transitivity for intuitionistic and interval-valued fuzzy preference relations and its application in deriving their priority vectors

Intuitionistic fuzzy preference relations (IFPRs) are used to deal with hesitation while interval-valued fuzzy preference relations (IVFPRs) are for uncertainty in multi-criteria decision making (MCDM). This article aims to explore the isomorphic multiplicative transitivity for IFPRs and IVFPRs, which builds the substantial relationship between hesitation and uncertainty in MCDM. To do that, the definition of the multiplicative transitivity property of IFPRs is established by combining the multiplication of intuitionistic fuzzy sets and Tanino's multiplicative transitivity property of fuzzy preference relations (FPRs). It is proved to be isomorphic to the multiplicative transitivity of IVFPRs derived via Zadeh's Extension Principle. The use of the multiplicative transitivity isomorphism is twofold: (1) to discover the substantial relationship between IFPRs and IVFPRs, which will bridge the gap between hesitation and uncertainty in MCDM problems; and (2) to strengthen the soundness of the multiplicative transitivity property of IFPRs and IVFPRs by supporting each other with two different reliable sources, respectively. Furthermore, based on the existing isomorphism, the concept of multiplicative consistency for IFPRs is defined through a strict mathematical process, and it is proved to satisfy the following several desirable properties: weak--transitivity, max-max--transitivity, and center-division--transitivity. A multiplicative consistency based multi-objective programming (MOP) model is investigated to derive the priority vector from an IFPR. This model has the advantage of not losing information as the priority vector representation coincides with that of the input information, which was not the case with existing methods where crisp priority vectors were derived as a consequence of modelling transitivity just for the intuitionistic membership function and not for the intuitionistic non-membership function. Finally, a numerical example concerning green supply selection is given to validate the efficiency and practicality of the proposed multiplicative consistency MOP model

New Trends in Neutrosophic Theory and Applications Volume II

Neutrosophic set has been derived from a new branch of philosophy, namely Neutrosophy. Neutrosophic set is capable of dealing with uncertainty, indeterminacy and inconsistent information. Neutrosophic set approaches are suitable to modeling problems with uncertainty, indeterminacy and inconsistent information in which human knowledge is necessary, and human evaluation is needed. Neutrosophic set theory was proposed in 1998 by Florentin Smarandache, who also developed the concept of single valued neutrosophic set, oriented towards real world scientific and engineering applications. Since then, the single valued neutrosophic set theory has been extensively studied in books and monographs introducing neutrosophic sets and its applications, by many authors around the world. Also, an international journal - Neutrosophic Sets and Systems started its journey in 2013. Single valued neutrosophic sets have found their way into several hybrid systems, such as neutrosophic soft set, rough neutrosophic set, neutrosophic bipolar set, neutrosophic expert set, rough bipolar neutrosophic set, neutrosophic hesitant fuzzy set, etc. Successful applications of single valued neutrosophic sets have been developed in multiple criteria and multiple attribute decision making. This second volume collects original research and application papers from different perspectives covering different areas of neutrosophic studies, such as decision making, graph theory, image processing, probability theory, topology, and some theoretical papers. This volume contains four sections: DECISION MAKING, NEUTROSOPHIC GRAPH THEORY, IMAGE PROCESSING, ALGEBRA AND OTHER PAPERS. First paper (Pu Ji, Peng-fei Cheng, Hongyu Zhang, Jianqiang Wang. Interval valued neutrosophic Bonferroni mean operators and the application in the selection of renewable energy) aims to construct selection approaches for renewable energy considering the interrelationships among criteria. To do that, Bonferroni mean (BM) and geometric BM (GBM) are employed

Managing Incomplete Preference Relations in Decision Making: A Review and Future Trends

In decision making, situations where all experts are able to efficiently express their preferences over all the available options are the exception rather than the rule. Indeed, the above scenario requires all experts to possess a precise or sufficient level of knowledge of the whole problem to tackle, including the ability to discriminate the degree up to which some options are better than others. These assumptions can be seen unrealistic in many decision making situations, especially those involving a large number of alternatives to choose from and/or conflicting and dynamic sources of information. Some methodologies widely adopted in these situations are to discard or to rate more negatively those experts that provide preferences with missing values. However, incomplete information is not equivalent to low quality information, and consequently these methodologies could lead to biased or even bad solutions since useful information might not being taken properly into account in the decision process. Therefore, alternative approaches to manage incomplete preference relations that estimates the missing information in decision making are desirable and possible. This paper presents and analyses methods and processes developed on this area towards the estimation of missing preferences in decision making, and highlights some areas for future research

Motion Planning for Optimal Information Gathering in Opportunistic Navigation Systems

Motion planning for optimal information gathering in an opportunistic navigation (OpNav) environment is considered. An OpNav environment can be thought of as a radio frequency signal landscape within which a receiver locates itself in space and time by extracting information from ambient signals of opportunity (SOPs). The receiver is assumed to draw only pseudorange-type observations from the SOPs, and such observations are fused through an estimator to produce an estimate of the receiver’s own states. Since not all SOP states in the OpNav environment may be known a priori, the receiver must estimate the unknown SOP states of interest simultaneously with its own states. In this work, the following problem is studied. A receiver with no a priori knowledge about its own states is dropped in an unknown, yet observable, OpNav environment. Assuming that the receiver can prescribe its own trajectory, what motion planning strategy should the receiver adopt in order to build a high-fidelity map of the OpNav signal landscape, while simultaneously localizing itself within this map in space and time? To answer this question, first, the minimum conditions under which the OpNav environment is fully observable are established, and the need for receiver maneuvering to achieve full observability is highlighted. Then, motivated by the fact that not all trajectories a receiver may take in the environment are equally beneficial from an information gathering point of view, a strategy for planning the motion of the receiver is proposed. The strategy is formulated in a coupled estimation and optimal control framework of a gradually identified system, where optimality is defined through various information-theoretic measures. Simulation results are presented to illustrate the improvements gained from adopting the proposed strategy over random and pre-defined receiver trajectories.Aerospace Engineering and Engineering Mechanic

New Challenges in Neutrosophic Theory and Applications

Neutrosophic theory has representatives on all continents and, therefore, it can be said to be a universal theory. On the other hand, according to the three volumes of “The Encyclopedia of Neutrosophic Researchers” (2016, 2018, 2019), plus numerous others not yet included in Encyclopedia book series, about 1200 researchers from 73 countries have applied both the neutrosophic theory and method. Neutrosophic theory was founded by Professor Florentin Smarandache in 1998; it constitutes further generalization of fuzzy and intuitionistic fuzzy theories. The key distinction between the neutrosophic set/logic and other types of sets/logics lies in the introduction of the degree of indeterminacy/neutrality (I) as an independent component in the neutrosophic set. Thus, neutrosophic theory involves the degree of membership-truth (T), the degree of indeterminacy (I), and the degree of non-membership-falsehood (F). In recent years, the field of neutrosophic set, logic, measure, probability and statistics, precalculus and calculus, etc., and their applications in multiple fields have been extended and applied in various fields, such as communication, management, and information technology. We believe that this book serves as useful guidance for learning about the current progress in neutrosophic theories. In total, 22 studies have been presented and reflect the call of the thematic vision. The contents of each study included in the volume are briefly described as follows. The first contribution, authored by Wadei Al-Omeri and Saeid Jafari, addresses the concept of generalized neutrosophic pre-closed sets and generalized neutrosophic pre-open sets in neutrosophic topological spaces. In the article “Design of Fuzzy Sampling Plan Using the Birnbaum-Saunders Distribution”, the authors Muhammad Zahir Khan, Muhammad Farid Khan, Muhammad Aslam, and Abdur Razzaque Mughal discuss the use of probability distribution function of Birnbaum–Saunders distribution as a proportion of defective items and the acceptance probability in a fuzzy environment. Further, the authors Derya Bakbak, Vakkas Uluc¸ay, and Memet S¸ahin present the “Neutrosophic Soft Expert Multiset and Their Application to Multiple Criteria Decision Making” together with several operations defined for them and their important algebraic properties. In “Neutrosophic Multigroups and Applications”, Vakkas Uluc¸ay and Memet S¸ahin propose an algebraic structure on neutrosophic multisets called neutrosophic multigroups, deriving their basic properties and giving some applications to group theory. Changxing Fan, Jun Ye, Sheng Feng, En Fan, and Keli Hu introduce the “Multi-Criteria Decision-Making Method Using Heronian Mean Operators under a Bipolar Neutrosophic Environment” and test the effectiveness of their new methods. Another decision-making study upon an everyday life issue which empowered us to organize the key objective of the industry developing is given in “Neutrosophic Cubic Einstein Hybrid Geometric Aggregation Operators with Application in Prioritization Using Multiple Attribute Decision-Making Method” written by Khaleed Alhazaymeh, Muhammad Gulistan, Majid Khan, and Seifedine Kadry