Oleg H Huseynov

Fuzzy logic-based generalized decision theory with imperfect information

The existing decision models have been successfully applied to solving many decision problems in management, business, economics and other fields, but nowadays arises a need to develop more realistic decision models. The main drawback of the existing utility theories starting from von Neumann-Moregnstern expected utility to the advanced non-expected models is that they are designed for laboratory examples with simple, well-defined gambles which do not adequately enough reflect real decision situations. In real-life decision making problems preferences are vague and decision-relevant information is imperfect as described in natural language (NL). Vagueness of preferences and imperfect decision relevant information require using suitable utility model which would be fundamentally different to the existing precise utility models. Precise utility models cannot reflect vagueness of preferences, vagueness of objective conditions and outcomes, imprecise beliefs, etc. The time has come for a new generation of decision theories. In this study, we propose a decision theory, which is capable to deal with vague preferences and imperfect information. The theory discussed here is based on a fuzzy-valued non-expected utility model representing linguistic preference relations and imprecise beliefs.

The Arithmetic of Z-Numbers: Theory and Applications

Real-world information is imperfect and is usually described in natural language (NL). Moreover, this information is often partially reliable and a degree of reliability is also expressed in NL. In view of this, the concept of a Z-number is a more adequate concept for the description of real-world information. The main critical problem that naturally arises in processing Z-numbers-based information is the computation with Z-numbers. Nowadays, there is no arithmetic of Z-numbers suggested in existing literature. This book is the first to present a comprehensive and self-contained theory of Z-arithmetic and its applications. Many of the concepts and techniques described in the book, with carefully worked-out examples, are original and appear in the literature for the first time.The book will be helpful for professionals, academics, managers and graduate students in fuzzy logic, decision sciences, artificial intelligence, mathematical economics, and computational economics. Readership: Researchers, academics, professionals and graduate students in fuzzy logic, decision sciences and artificial intelligence.

DECISION THEORY WITH IMPRECISE PROBABILITIES

There is an extensive literature on decision making under uncertainty. Unfortunately, up to date there are no valid decision principles. Experimental evidence has repeatedly shown that widely used principle of maximization of expected utility has serious shortcomings. Utility function and nonadditive measures used in nonexpected utility models are mainly considered as real-valued functions whereas in reality decision-relevant information is imprecise and therefore is described in natural language. This applies, in particular, to imprecise probabilities expressed by terms such as likely, unlikely, probable, etc. The principal objective of the paper is the development of computationally effective methods of decision making with imprecise probabilities. We present representation theorems for a nonexpected fuzzy utility function under imprecise probabilities. We develop an effective decision theory when the environment of fuzzy events, fuzzy states, fuzzy relations and fuzzy constraints are characterized by imprecise probabilities. The suggested methodology is applied for a real-life decision-making problem.

Z-Number-Based Linear Programming

Linear programming (LP) is the operations research technique frequently used in the fields of science, economics, business, management science, and engineering. Although it is investigated and applied for more than six decades, and LP models with different level of generalization of information about parameters including models with interval, fuzzy, generalized fuzzy, and random numbers are considered, until now there is no approach to account for reliability of information within the framework of LP.

Decision Theory with Imperfect Information

Every day decision making in complex human-centric systems are characterized by imperfect decision-relevant information. The principal problems with the existing decision theories are that they do not have capability to deal with situations in which probabilities and events are imprecise. In this book, we describe a new theory of decision making with imperfect information. The aim is to shift the foundation of decision analysis and economic behavior from the realm bivalent logic to the realm fuzzy logic and Z-restriction, from external modeling of behavioral decisions to the framework of combined states. This book will be helpful for professionals, academics, managers and graduate students in fuzzy logic, decision sciences, artificial intelligence, mathematical economics, and computational economics. Readership: Professionals, academics, managers and graduate students in fuzzy logic, decision sciences, artificial intelligence, mathematical economics, and computational economics.

The general theory of decisions

The majority of the existing decision theories, starting from von Neumann and Morgenstern Expected Utility theory, are based on sound mathematical background and yielded good results. However, they are developed for solving particular decision situations. Consequently, nowadays there is no theory to unite main directions of decision analysis. The existing main theories suffer from several important disadvantages: (i) use of numerical techniques which contrast with real-world imperfect information; (ii) the assumptions of well-structured knowledge about future objective conditions; (iii) the use of probability measures whereas real-world probabilities are imprecise; (iv) the use binary logic-based preference relations, whereas real preferences may be vague; (v) no account for the fact that a human being reasons with linguistic description of information; (vi) parametrical modeling of behavioral determinants without account for interaction; (vii) missing partial reliability of real decision-relevant information. Thus, it becomes needed to develop a general theory of decisions that would be free of the limitations outlined above. In this paper, we propose the fundamentals of the new general theory of decisions which is based on complex consideration of imperfect decision-relevant information issues and behavioral aspects. We illustrate that the existing theories including Expected Utility of von Neumann and Morgenstern, Prospect Theory, Choquet Expected Utility, Cumulative Prospect Theory and other theories are special cases of the suggested general theory of decisions. We provide axioms and principles, the corresponding mathematical methodologies of decision analysis and auxiliary formal techniques. The application of the suggested theory is illustrated with the aid of a benchmark complex decision problem.

Fuzzy optimality based decision making under imperfect information without utility

Abstract In the realm of decision making under uncertainty, the general approach is the use of the utility theories. The main disadvantage of this approach is that it is based on an evaluation of a vector-valued alternative by means of a scalar-valued quantity. This transformation is counterintuitive and leads to loss of information. The latter is related to restrictive assumptions on preferences underlying utility models like independence, completeness, transitivity etc. Relaxation of these assumptions results into more adequate but less tractable models. In contrast, humans conduct direct comparison of alternatives as vectors of attributes’ values and don’t use artificial scalar values. Although vector-valued utility function-based methods exist, a fundamental axiomatic theory is absent and the problem of a direct comparison of vectors remains a challenge with a wide scope of research and applications. In the realm of multicriteria decision making there exist approaches like TOPSIS and AHP to various extent utilizing components-wise comparison of vectors. Basic principle of such comparison is the Pareto optimality which is based on a counterintuitive assumption that all alternatives within a Pareto optimal set are considered equally optimal. The above mentioned mandates necessity to develop new decision approaches based on direct comparison of vector-valued alternatives. In this paper we suggest a fuzzy Pareto optimality (FPO) based approach to decision making with fuzzy probabilities representing linguistic decision-relevant information. We use FPO concept to differentiate “more optimal” solutions from “less optimal” solutions. This is intuitive, especially when dealing with imperfect information. An example is solved to show the validity of the suggested ideas.

The arithmetic of continuous Z-numbers

In order to deal with imprecision and partial reliability of real-world information, Prof. Zadeh suggested the concept of a Z-number Z=(A, B), as an ordered pair of continuous fuzzy numbers A and B. The first describes a linguistic value, and the second one is the associated reliability. Unfortunately, up to day there is no works devoted to arithmetic of continuous Z-numbers in existence. An original formulation of operations over continuous Z-numbers proposed by Zadeh includes complex non-linear variational problems. We propose an alternative approach which has a better computational complexity and accuracy tradeoff. The proposed approach is based on linear programming and other simple optimization problems. We developed basic arithmetic operations such as addition, subtraction, multiplication and division, and some algebraic operations as maximum, minimum, square and square root of continuous Z-numbers. Vast compendium of examples shows validity of the suggested approach.

BEHAVIORAL DECISION MAKING WITH COMBINED STATES UNDER IMPERFECT INFORMATION

Behavioral decision making is an area of multidisciplinary research attracting growing interest of scientists and practitioners, economists, and business people. A wide spectrum of successful theories is present now, including Prospect theory, multiple priors models, studies on altruism, trust and fairness. However, these theories are developed for precise and complete information, whereas real information concerning a decision makers (DM) behavior and environment is imperfect, qualitative, and, as a result, often described in natural language (NL). We suggest an approach based on modeling a DMs behavior by a set of states. Each state represents a certain principal behavior. In our approach, states of nature and DMs states constitute a single space of combined states. For formalizing relevant information described in NL, we use fuzzy set theory. The utility model is based on Choquet-like integration over combined states. The investigations show that Expected Utility, Choquet Expected Utility and Cumulative Prospect Theory are special cases of the suggested approach. We apply the suggested approach to solving a benchmark and a real-life decision problem. The obtained results show validity of the suggested approach.

Decision Making with Second Order Information Granules

Decision-making under uncertainty has evolved into a mature field. However, in most parts of the existing decision theory, one assumes decision makers have complete decision-relevant information. The standard framework is not capable to deal with partial or fuzzy information, whereas, in reality, decision-relevant information about outcomes, probabilities, preferences etc is inherently imprecise and as such described in natural language (NL). Nowadays, there is no decision theory with second-order uncertainty in existence albeit real-world uncertainties fall into this category. This applies, in particular, to imprecise probabilities expressed by terms such as likely, unlikely, probable, usually etc. We call such imprecise evaluations second-order information granules.