Chapter Twenty - Social Choice with Fuzzy Preferences

Granular structure plays a very important role in the model construction, theoretical analysis and algorithm design of a granular computing method. The granular structures of classical rough sets and fuzzy rough sets have been proven to be clear. In classical rough set theory, equivalence classes are basic granules, and the lower and upper approximations of a set can be computed by those basic granules. In the theory of fuzzy rough set, granular fuzzy sets can be used to describe the lower and upper approximations of a fuzzy set. This paper discusses the granular structure of type-2 fuzzy rough sets over two universes. Definitions of type-2 fuzzy rough sets over two universes are given based on a wavy-slice representation of type-2 fuzzy sets. Two granular type-2 fuzzy sets are deduced and then proven to be basic granules of type-2 fuzzy rough sets over two universes. Then, the properties of lower and upper approximation operators and these two granular type-2 fuzzy sets are investigated. At last, several examples are given to show the applications of type-2 fuzzy rough sets over two universes.

This chapter formally introduces type-1 fuzzy sets and fuzzy logic. It is the backbone for Chap. 3 and provides the foundation upon which type-2 fuzzy sets and systems are built in later chapters. Its coverage includes: crisp sets, type-1 fuzzy sets and associated concepts [including a short biography of Prof. Zadeh (the father of fuzzy sets and fuzzy logic)], type-1 fuzzy set defined, linguistic variables, returning to linguistic variables from a numerical value of a membership function, set theoretic operations for crisp and type-1 fuzzy sets, crisp and fuzzy relations and compositionson the same or different product spaces , compositions of a type-1 fuzzy set with a type-1 fuzzy relation, hedges, the Extension Principle (which is about functions of fuzzy sets), α-cuts (which are a powerful way to represent a type-1 fuzzy set in terms of intervals), functions of type-1 fuzzy sets computed by using α-cuts, multivariable membership functions and Cartesian products, crisp logic, going from crisp logic to fuzzy logic, Mamdani (engineering) implications, some final remarks, and an appendix about properties/laws of type-1 fuzzy sets. 35 examples are used to illustrate this chapter’s important concepts.

In this paper we first consider the problem of extending fuzzy (weak and strict) preference relations, represented by fuzzy preorders on a set to a fuzzy preferences on subsets, and we characterise different possibilities. Based on their properties, we then semantically define and axiomatize several two-tiered graded modal logics to reason about the corresponding different notions of fuzzy preferences.

Fuzzy types: a framework for handling uncertainty about types of objects

Judgment aggregation is a recent theory that combines aggregation problems previously studied by social choice theory with logic. Social choice theory is a vast subject, including not only the study of preference aggregation and voting theory but also topics like social welfare and justice. Given the tight links between judgment aggregation and preference aggregation, in this first chapter we give a concise survey on some historical aspects of preference aggregation and then introduce and motivate the more recent field of judgment aggregation. The chapter builds on [Sen99, Sen86, Bla58] for the historical overview on social choice theory, and on [KS93, Kor92] for the informal introduction to judgment aggregation.

In this paper a theoretical and practical comparative study is proposed, between types 1 and 2 fuzzy systems for the monitoring of a gas turbine process. Firstly, the type 1 and type 2 fuzzy sets are presented followed by the description of the two fuzzy systems, to understand the differences between there inference systems, then a theoretical comparative study is presented, and finally the contribution of this work in the practical part by the application of the two fuzzy control approaches type-1 and type-2 in a fuel valve positioning of a gas turbine process, the obtained results shows the advantage of the used time in fuzzy control type-2, which is a very important factor in industrial applications.

Recently, fuzzy linguistic variables have gained a great deal of attention from researchers in many decision-making problems. In this problems, type-2 fuzzy sets have been used to better cover linguistic data uncertainty. However, many of research works in this regard have been performed in type-2 fuzzy domain and in static mode. Since the decision making problems in the real world usually fluctuate over time, so it need to use decision making models in multi period of time. In the present research, a Multi-Period (Dynamic) Multi-Attribute Group Decision-Making (MPMADM) method is presented based on type-2 fuzzy sets where decision-making attributes are first expressed in linguistic terms and then incorporated, as interval type-2 fuzzy numbers, into problem solving where a new integrating operator called Multi-Period Trapezoidal Interval Type-2 Fuzzy Number Weighted Arithmetic averaging (MPTIT2FNWA) is defined on type-2 interval fuzzy numbers to integrate decision-making information in multiple periods of time. Once finished with explaining the proposed method, a numerical example is given to evaluate the proposed method in terms of effectiveness and applicability, with the results compared to those of other methods.

Computational social choice is a new interdisciplinary research area at the intersection of social choice theory and computer science that studies computational aspects of social choice mechanisms as well as applications of social choice ideas in the design and analysis of computing systems. Research on computational social choice now encompasses a variety of topics, including the algorithmic aspects of voting rules, computational barriers to strategic behaviour, collective decision-making in multi-agent systems, preference elicitation and communication issues in voting, fair division, computational aspects of weighted voting games, collective decision-making in combinatorial domains, logic-based formalisms for social choice problems, belief and judgement aggregation, and social software. The aim of this special issue is to build on the spectacular success of two workshops on computational social choice (COMSOC’06, Amsterdam and COMSOC’08, Liverpool), and provide a forum for reseachers in the area to present their ideas on a variety of topics. This special issue is composed of nine research articles and a short note that span a fairly wide range of issues, including voting, mechanism design, judgment aggregation, and fairness, and a similarly wide range of techniques, such as logic, experiments, and computational complexity. The paper by Agotnes, van der Hoek and Wooldridge defines a new modal logic for representing and reasoning about preference and judgment aggregation. The authors show that their logic can be used to express a variety of concepts and results from the social choice domain, including majority voting, the discursive paradox, Arrow’s theorem and Condorcet’s paradox. Baharad, Goldberger, Koppel and Nitzan propose a maximum likelihood estimation-based approach for aggregating voters’ opinions over several issues. They argue that their method is vastly superior to simple majority voting, and use both analytical and empirical evaluation to support their argument. Balan, Richards and Luke deal with the problem of selecting a sequence of actions when the actions have multiple beneficiaries. Their goal is to select the actions according to a lexicographic fairness criterion. The explore the complexity of this problem for both finite and infinite sequences of actions and present both algorithms and intractability results for this problem. The paper by Caminada and Pigozzi applies argumentation theory to judgment aggregation, and proposes aggregation operators satisfying not only the usual property of collective rationality, but another highly desirable property,

Since Lotfi A. Zadeh introduced the concept of fuzzy sets in 1965, many authors have devoted their efforts to the study of these new sets, both from a theoretical and applied point of view. Fuzzy sets were later extended in order to get more adequate and flexible models of inference processes, where uncertainty, imprecision or vagueness is present. Type 2 fuzzy sets comprise one of such extensions. In this paper, we introduce and study an extension of the fuzzy numbers (type 1), the type 2 generalized fuzzy numbers and type 2 fuzzy numbers. Moreover, we also define a partial order on these sets, which extends into these sets the usual order on real numbers, which undoubtedly becomes a new option to be taken into account in the existing total preorders for ranking interval type 2 fuzzy numbers, which are a subset of type 2 generalized fuzzy numbers.

Nowadays, most people can get enough energy to maintain one-day activity, while few people know whether they eat healthily or not. It is quite important to analyze nutritional facts of foods eaten for those who are losing weight or suffering chronic diseases such as diabetes. However, diet is a problem with a high uncertainty, and it is widely pointed out that classical ontology is not sufficient to deal with imprecise and vague knowledge for some real-world applications like diet. On the other hand, a fuzzy ontology can effectively help handle and process uncertain data and knowledge. This chapter proposes a type-2 fuzzy set and fuzzy ontology for diet application and uses the type-2 fuzzy markup language (T2FML) to describe the knowledge base and rule base of the diet, including ingredients and the contained servings of six food categories of some common foods in Taiwan. The experimental results show that type-2 fuzzy logic system (FLS) performs better than type-1 FLS, proving that type-2 FLS can provide a powerful paradigm to handle the high level of uncertainties present in diet.

1. Introductory Sections.- Multiperson Decision Making: a Selective Review.- Fuzzy Set Theory as a Theory of Vagueness.- Vague Notions in the Theory of Voting.- 2. General Issues Related to Decision Making under Fuzziness.- Aggregation of Possibility Measures.- Modelling Valued Preference Relations.- Revealed Fuzzy Preferences.- Categories of Fuzzy Relations in Decision Making.- Determination and Interpretation of the Fuzzy Utility of an Act in an Uncertain Environment.- Extending Aggregation Operators for Multicriteria Decision Making.- Ranking Alternatives by Weak Transitivity Relations.- Calculating the Mean Knowledge Representation from Multiple Experts.- An Approach to Customized End-User Views in Multi-User Information Retrieval Systems.- 3. Group Decision Making under Fuzziness.- Means and Social Welfare Functions in Fuzzy Binary Relation Spaces.- Aggregation of Fuzzy Preferences.- Single - Peakedness in Weighted Aggregation of Fuzzy Opinions in a Fuzzy Group.- On Group Decision Making under Fuzzy Preferences.- Group Decision Making with Fuzzy and Non-Fuzzy Evaluations.- On Construction of the Fuzzy Multiattribute Risk Function for Group Decision Making.- Consensus Measures for Qualitative Order Relations.- On a Consensus Measure in a Group MCDM Problem.- Voting Procedures with a priori Incomplete Individual Profiles.- 4. Team Decision Making under Fuzziness.- A Team Decision Making Model for Distributed Problem Solving.- Evidential Teams.- 5. Fuzzy Games.- Fuzzy Goals and Sets of Choices in Two-Person Games.- Playing Matrix Games Defined by Linguistic Labels.- Fuzzy Convexity and Peripherial Core of an Exchange Economy Represented as a Fuzzy Game.- Fuzzy Sequencing Games.

There are alternative ways of decomposing a given fuzzy weak preference relation into its antisymmetric and symmetric components. In this paper I have provided support to one among these alternative specifications. It is shown that on this specification the fuzzy analogue of the General Possibility Theorem is valid even when the transitivity restrictions on the individual and the social preference relations are relatively weak. In the special case where the individual preference relations are exact but the social preference relation is permitted to be fuzzy it is possible to distinguish between different degrees of power of the dictator. This power increases with the strength of the transitivity requirement.

The main aim of this paper is to introduce the definition of intuitionistic fuzzy set and operations on it that will prove fruitful to conclude to a solution on problems of real life and how fuzzy logic and Graphics Processing unit can be used together to solve different domain problems for faster responses. Various comprehensive survey of Type-2 fuzzy set and the intuitionistic fuzzy set theories have been described here. IFS is nothing but a simple extension of the fuzzy set defined in the domain of discourse. Graphics Processing Unit (GPU) has been used to enhance the speed of execution. The intuitionistic fuzzy set is utilized as a proper element for representing the degree of hesitation concerning both the non-membership and membership degrees of an element of a set. Recent trends show how GPU is used for various computational applications that are run parallel to reduce overall execution time. Both the fuzzy domains seek application in tackling of the situation of vagueness with their respective characteristic feature sets to the fields concerned to different engineering domains of interdisciplines. This paper highlights the overview of the two eminent fuzzy categories like Fuzzy type 2 and intuitionistic fuzzy theory along with their application on greater scale.

Emerging trends in soft set theory and related topics.

Since L. Zadeh proposed the concept of a fuzzy set in 1965 the relationships between probability theory and possibility theory have been discussed. Both theories seem to be similar in the sense that they both are concerned with some type of uncertainty and both use the [0,1] interval for their measures as the range of their respective functions. (At least as long as one considers normalized fuzzy sets only!) Other uncertainty measures, which were already mentioned in chapter 4, also focus on uncertainty and could therefore be included in such a discussion. The comparison between probability theory and fuzzy set theory is difficult primarily for two reasons: 1. The comparison could be made on very different levels, that is, mathematically, semantically, linguistically, and so on. 2. Fuzzy set theory is not or is no longer a uniquely defined mathematical structure, such as Boolean algebra or dual logic. It is rather a very general family of theories (consider, for instance, all the possible operations which were already defined in chapter 3 or of different types of membership functions). In this respect, fuzzy set theory could rather be compared with the different existing theories of multivalued logic.