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Über dieses Buch

This book offers a multifaceted perspective on fuzzy set theory, discussing its developments over the last 50 years. It reports on all types of fuzzy sets, from ordinary to hesitant fuzzy sets, with each one explained by its own developers, authoritative scientists well known for their previous works. Highlighting recent theorems and proofs, the book also explores how fuzzy set theory has come to be extensively used in almost all branches of science, including the health sciences, decision science, earth science and the social sciences alike. It presents a wealth of real-world sample applications, from routing problem to robotics, and from agriculture to engineering. By offering a comprehensive, timely and detailed portrait of the field, the book represents an excellent reference guide for researchers, lecturers and postgraduate students pursuing research on new fuzzy set extensions.



Fuzzy Sets Theory: From the Past to the Future


The Emergence of Fuzzy Sets: A Historical Perspective

This paper tries to suggest some reasons why fuzzy set theory came to life 50 years ago by pointing out the existence of streams of thought in the first half of the XXth century in logic, linguistics and philosophy, that paved the way to the idea of moving away from the Boolean framework, through the proposal of many-valued logics and the study of the vagueness phenomenon in natural languages. The founding paper in fuzzy set theory can be viewed as the crystallization of such ideas inside the engineering arena. Then we stress the point that this publication in 1965 was followed by several other seminal papers in the subsequent 15 years, regarding classification, ordering and similarity, systems science, decision-making, uncertainty management and approximate reasoning. The continued effort by Zadeh to apply fuzzy sets to the basic notions of a number of disciplines in computer and information sciences proved crucial in the diffusion of this concept from mathematical sciences to industrial applications.
Didier Dubois, Henri Prade

Fuzzy Decision Making: Its Pioneers and Supportive Environment

Fuzzy decision making is the collection of single or multicriteria techniques aiming at selecting the best alternative in case of imprecise, incomplete, and vague data. This chapter reviews the fuzzy decision making literature and summarizes the review results by tabular and graphical illustrations. The classification is based on the new extensions of fuzzy sets: Intuitionistic, hesitant, and type-2 fuzzy sets. Later, the media publishing fuzzy decision making papers, journals, books, conferences, and societies are summarized. Finally, fuzzy decision making examples are given for ordinary, intuitionistic, hesitant, and type-2 fuzzy sets.
Cengiz Kahraman, Sezi Çevik Onar, Başar Öztayşi

Mathematics of Fuzzy Sets


Mathematics of Intuitionistic Fuzzy Sets

Short firsthand remarks on the history and theory of Intuitionistic Fuzzy Sets (IFSs) are given. Influences of other areas of mathematics for development of the IFSs theory are discussed. On the basis of results in IFSs theory, some ideas for development of other mathematical areas are offered.
Krassimir Atanassov

Some Comments on Ordinary Reasoning with Fuzzy Sets

The main goal of Computing with Words is essentially a calculation allowing to automate a part of the reasoning done thanks to the natural language. Fuzzy Logic is the main tool to perform this calculation because it is be able to represent the most common kind of predicates in natural language, graded predicates, in terms of functions, and to calculate with them. However there is still not an adequate framework to perform this task, commonly referred to as commonsense reasoning. This chapter proposes a general framework to model a part of this type of reasoning. The fundamental fact of this framework is its ability to adequately represent noncontradiction, the minimum condition for considering a reasoning as valid. Initially, the characteristics of the commonsense reasoning are analyzed, and a model for the crisp case is shown. After that the more general case in which graded predicates are taken under consideration is studied.
Enric Trillas, Adolfo R. de Soto

A Review of Hesitant Fuzzy Sets: Quantitative and Qualitative Extensions

Since the concept of fuzzy set was introduced, different extensions and generalizations have been proposed to manage the uncertainty in different problems. This chapter is focused in a recent extension so-called hesitant fuzzy set. Many researchers have paid attention on it and have proposed different extensions both in quantitative and qualitative contexts. Several concepts, basic operations and its extensions are revised in this chapter.
Rosa M. Rodríguez, Luis Martínez, Francisco Herrera, Vicenç Torra

Type-1 to Type-n Fuzzy Logic and Systems

In this chapter, the motivation for using fuzzy systems, the mathematical concepts of type-1 to type-n fuzzy sets, logic, and systems as well as their applications in solving real world problems are presented.
M. H. Fazel Zarandi, R. Gamasaee, O. Castillo

Fuzzy Sets in Branches of Science


Fuzzy Sets in Earth and Space Sciences

Earth science refers to the field of science dealing with planet Earth while space science pertains several scientific disciplines studying the upper atmosphere, space, and celestial bodies rather than Earth. The fuzzy set theory is one of the tools that has been recently used in the earth and space sciences. In this chapter, we review and analyze the papers utilizing fuzzy logic in earth and space science problems from Scopus database. The graphical and tabular illustrations are presented for the subject areas, publication years and sources of the papers on earth and space sciences.
Irem Otay, Cengiz Kahraman

Fuzzy Sets and Fuzzy Logic in the Human Sciences

The development of fuzzy set theory and fuzzy logic provided an opportunity for the human sciences to incorporate a mathematical framework with attractive properties. The potential applications include using fuzzy set theory as a descriptive model of how people treat categorical concepts, employing it as a prescriptive framework for “rational” treatment of such concepts, and as a basis for analysing graded membership response data from experiments and surveys. However, half a century later this opportunity still has not been fully grasped. This chapter surveys the history of fuzzy set applications in the human sciences, and then elaborates the possible reasons why fuzzy set concepts have been relatively under-utilized therein.
Michael Smithson

Fuzzy Entropy Used for Predictive Analytics

Process interruptions in (very) large production systems are difficult to deal with. Modern processes are highly automated; data is collected with sensor technology that forms a big data context and offers challenges to identify coming failures from the very large sets of data. The sensors collect huge amounts of data but the failure events are few and infrequent and hard to find (and even harder to predict). In this article, our goal is to develop models for predictive maintenance in a big data environment. The purpose of feature selection in the context of predictive maintenance is to identify a small set of process diagnostics that are sufficient to predict future failures. We apply interval-valued fuzzy sets and various entropy measures defined on them to perform feature selection on process diagnostics. We show how these models can be utilized as the basis of decision support systems in process industries to aid predictive maintenance.
Christer Carlsson, Markku Heikkilä, József Mezei

Fuzzy Sets in Agriculture

Agricultural modeling and management are complex conceptual processes, where a large number of variables are taken into consideration and interact for system analysis and decision making. Most of the processes in the agricultural sector include the uncertainty, ambiguity, incomplete information and human intuition characteristics. These processes are not only constrained by their environment (e.g., market, climate, seasons, consumer choices), but they are also highly influenced by human factors (stakeholders’ perceptions). Fuzzy sets are able to manage and represent uncertainty, assure that the incomplete information is valued and provide solutions to issues which are crucial in agriculture like fertilization, land degradation, soil erosion and climate variability during planting material selection in physiological analysis. Fuzzy sets have gained constantly increasing research interest in the last twenty years and have found great applicability in the agricultural domain, helping farmers to take right decisions for their cultivated.
Elpiniki I. Papageorgiou, Konstantinos Kokkinos, Zoumpoulia Dikopoulou

Applications of Fuzzy Sets


Solving a Multiobjective Truck and Trailer Routing Problem with Fuzzy Constraints

The Truck and Trailer Routing Problem uses trucks pulling trailers as a distinctive feature of the Vehicle Routing Problem. Recently, this problem has been treated considering the capacity constraints as fuzzy. This situation means that the decision maker admits the violation of these constraints according to a value of tolerance. This relaxation can generate a set of solutions with very low costs but its non-fulfillment grade of the capacity constraints can be high and vice versa. This fuzzy variant is generalized in this work from a multiobjective approach by incorporating an objective to minimize the violation of constraints. We present and discuss the computational experiments carried out to solve the multiobjective Truck and Trailer Routing Problem with fuzzy constraint using benchmark instances with sizes ranging from 50 to 199 customers.
Isis Torres, Alejandro Rosete, Carlos Cruz, José L. Verdegay

Health Service Network Design Under Epistemic Uncertainty

If a health system wants to achieve its strategic goal known as “reducing health inequalities”, making health services available and accessible to all people is an essential prerequisite. Health service network design (HSND) is known as one of the most critical strategic decisions that affects performance of health systems to the great extent. Important decisions such as location of health service providers (i.e. clinics, hospitals, etc.), allocation of patient zones to health service providers and optimal designing of patients flow via the network are some of the main strategic and tactical decisions that should be made when configuring a health service network. On the other hand, coping with uncertainty in data is an inseparable part of strategic and tactical problems. More specifically, the complex structure of health service networks alongside the volatile environment surrounding the health systems would impose a higher degree of uncertainty to the decision makers and health network designers. Among different methods to cope with uncertainty, possibilistic programming approaches are well-applied methods that can handle epistemic uncertainty in parameters.
Mohammad Mousazadeh, S. Ali Torabi, Mir Saman Pishvaee

Robotics and Control Systems

Robots are of those intelligent systems created to do a wide range of activities with the aim of human aid and productivity improvement. Besides, many different fields of studies such as engineering, healthcare, computer science, mathematics and management are involved in order to increase the efficiency and effectiveness of robots. Generally speaking, robotics and control systems is a branch of engineering science that deals with all aspects of robot’s design, operation and control. More precisely, the concept of control in this paper is knowing the techniques required for programming robot’s activities such as its physical movements, rotations, decisions and planning. In addition to mathematical modeling optimization and scheduling, there are a lot of control theory based approaches dealing with physical movement control of the robot at every moment of time. Due to the uncertainties, fuzzy set theory, applicable for all control techniques, is extensively used for robots. The role of fuzzy modeling becomes more evident when one can include human expertise and knowledge via fuzzy rules in the control system. Without loss of generality, this paper presents fuzzy control techniques as well as fuzzy mathematical scheduling model for an m-machine robotic cell with one manipulator robot. Furthermore, it proposes an integrated fuzzy robotic control system, in which the fuzzy optimization model is solved at every predetermined period of time such as beginning of shifts or days, etc. Then, based on the solutions obtained, input parameters and unpredictable disturbances, the autonomous fuzzy control is executed continuously. These two modules transfer information and feedback to each other via an intermediate collaborative module. The explanations are supported via an example.
M. H. Fazel Zarandi, H. Mosadegh

Fuzzy Sets in the Evaluation of Socio-Ecological Systems: An Interval-Valued Intuitionistic Fuzzy Multi-criteria Approach

In recent days, the use of the fuzzy set theory to deal with social complexity has become more attractive as a research area to academics who wish to contribute to sustainable development. The socio-ecological systems are one of the well-known sub-disciplines of social science which is very critical issue to maintain sustainability and this concept basically concerns with the human-environment interactions. These systems involve various stakeholders with different levels of knowledge and experience from diverse social platforms. The basic characteristic of these systems is identified as having high level of uncertainty and incomplete information. The main purpose of this chapter is to demonstrate how to incorporate fuzzy sets theory into social sciences. In this chapter, an illustrative example which consists of a wide variety of social actors is used to evaluate sustainable management options utilizing the extended technique for order preference by similarity to ideal solution (TOPSIS) method for interval valued intuitionistic fuzzy multi-criteria group decision making.
Beyzanur Çayır Ervural, Bilal Ervural, Cengiz Kahraman

A Survey on Models and Methods for Solving Fuzzy Linear Programming Problems

Fuzzy Linear Programming (FLP), as one of the main branches of operation research, is concerned with the optimal allocation of limited resources to several competing activities on the basis of given criteria of optimality in fuzzy environment. Numerous researchers have studied various properties of FLP problems and proposed different approaches for solving them. This work presents a survey on models and methods for solving FLP problems. The solution approaches are divided into four areas: (1) Linear Programming (LP) problems with fuzzy inequalities and crisp objective function, (2) LP problems with crisp inequalities and fuzzy objective function, (3) LP problems with fuzzy inequalities and fuzzy objective function and (4) LP problems with fuzzy parameters. In the first area, the imprecise right-hand-side of the constraints is specified with a fuzzy set and a monotone membership function expressing the individual satisfaction of the decision maker. In the second area, a fuzzy goal and corresponding tolerance is defined for each coefficient of decision variables in the objective function. In the third area, both the coefficients of decision variables in the objective function and the right-hand-side of the constraints are specified by fuzzy goals and corresponding tolerances. In the fourth area, some or all parameters of the LP problem are represented in terms of fuzzy numbers. In this contribution, some of the most common models and procedures for solving FLP problems are analyzed. The solution approaches are illustrated with numerical examples.
Ali Ebrahimnejad, José L. Verdegay

Applications of Fuzzy Mathematical Programming Approaches in Supply Chain Planning Problems

Supply chain planning includes numerous decision problems over strategic (i.e. long-term), tactical (i.e. mid-term) and operational (i.e. short-term) planning horizons in a supply chain. As most of supply chain planning problems deal with decision making in real world while configuring future situations, relevant data should be predicted and described for multiple time periods in the future. Such prediction and description involve imprecision and vagueness due to errors and absence of sharp boundaries in the subjective data and/or insufficient or unreliable objective data. If the uncertainty in supply chain planning problems is to be neglected by the decision maker, the plausible performance of supply chain in future conditions will be in doubt. This is why considerable body of the recent literature account for uncertainty through applying different uncertainty programming approaches with respect to the nature of uncertainty. This chapter aims to provide useful and updated information about different sources and types of uncertainty in supply chain planning problems and the strategies used to confront with uncertainty in such problems. A hyper methodological framework is proposed to cope with uncertainty in supply chain planning problems. Also, among the different uncertainty programming approaches, various fuzzy mathematical programming methods extended in the recent literature are introduced and a number of them are elaborated. Finally, a useful case study is illustrated to present the practicality of fuzzy programming methods in the area of supply chain planning.
Mohammad Javad Naderi, Mir Saman Pishvaee, Seyed Ali Torabi


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