Real-life decisions are usually made in the state of uncertainty such as randomness and fuzziness. How do we model optimization problems in uncertain environments? How do we solve these models? In order to answer these questions, this book provides a self-contained, comprehensive and up-to-date presentation of uncertain programming theory, including numerous modeling ideas, hybrid intelligent algorithms, and applications in system reliability design, project scheduling problem, vehicle routing problem, facility location problem, and machine scheduling problem. Researchers, practitioners and students in operations research, management science, information science, system science, and engineering will find this work a stimulating and useful reference.

This book provides comprehensive coverage of uncertain programming theory, including numerous modeling ideas, hybrid intelligent algorithms, and applications in system reliability design, vehicle routing problem, and machine scheduling problem.

This rapidly developing field encompasses many disciplines including operations research, mathematics, and probability. Conversely, it is being applied in a wide variety of subjects ranging from agriculture to financial planning and from industrial engineering to computer networks. This textbook provides a first course in stochastic programming suitable for students with a basic knowledge of linear programming, elementary analysis, and probability. The authors present a broad overview of the main themes and methods of the subject, thus helping students develop an intuition for how to model uncertainty into mathematical problems, what uncertainty changes bring to the decision process, and what techniques help to manage uncertainty in solving the problems. The early chapters introduce some worked examples of stochastic programming, demonstrate how a stochastic model is formally built, develop the properties of stochastic programs and the basic solution techniques used to solve them. The book then goes on to cover approximation and sampling techniques and is rounded off by an in-depth case study. A well-paced and wide-ranging introduction to this subject.

An up-to-date, authoritative, comprehensive look at optimization theory in uncertain environments Real-life management decisions, such as buy/sell decisions in the stock market, are almost always made in uncertain environments. Is it possible to make model decision problems to fit these circumstances? Once constructed, can these models be solved? In Uncertain Programming, Baoding Liu answers both of these questions in the affirmative and goes on to lay a solid foundation for optimization in generally uncertain environments. Uncertain Programming describes the basic concepts of mathematical programming, provides a genetic algorithm for optimization problems, and introduces the techniques of stochastic and fuzzy simulation. After examining some basic results of expected value models, the book moves on to explore chance-constrained programming with stochastic parameters and illustrate applications of chance-constrained programming models. Dr. Liu discusses dependent-chance programming in stochastic environments and extends both chance-constrained and dependent-chance programming from stochastic to fuzzy environments. He then constructs a theoretical framework for fuzzy programming with fuzzy rather than crisp decisions. This remarkable and revolutionary book: * Lays a foundation for optimization theory in uncertain environments * Provides a unifying principle for dealing with stochastic and fuzzy programming * Incorporates the most recent developments in the field * Emphasizes modeling ideas, evolutionary computation, and applications of uncertain programming Uncertain Programming is a reliable, authoritative, and eye-opening guide for researchers and engineers in operations research, management science, business management, information and systems science, and computer science.

Uncertainty theory is a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms. Uncertainty is any concept that satisfies the axioms of uncertainty theory. Thus uncertainty is neither randomness nor fuzziness. It is also known from some surveys that a lot of phenomena do behave like uncertainty. How do we model uncertainty? How do we use uncertainty theory? In order to answer these questions, this book provides a self-contained, comprehensive and up-to-date presentation of uncertainty theory, including uncertain programming, uncertain risk analysis, uncertain reliability analysis, uncertain process, uncertain calculus, uncertain differential equation, uncertain logic, uncertain entailment, and uncertain inference. Mathematicians, researchers, engineers, designers, and students in the field of mathematics, information science, operations research, system science, industrial engineering, computer science, artificial intelligence, finance, control, and management science will find this work a stimulating and useful reference.

This book provides a self-contained, comprehensive and up-to-date presentation of uncertainty theory. The purpose is to equip the readers with an axiomatic approach to deal with uncertainty. For this new edition the entire text has been totally rewritten. The chapters on chance theory and uncertainty theory are completely new. Mathematicians, researchers, engineers, designers, and students will find this work a stimulating and useful reference.

This book is intended to present the milestones in the progression of uncertain Data envelopment analysis (DEA). Chapter 1 gives some basic introduction to uncertain theories, including probability theory, credibility theory, uncertainty theory and chance theory. Chapter 2 presents a comprehensive review and discussion of basic DEA models. The stochastic DEA is introduced in Chapter 3, in which the inputs and outputs are assumed to be random variables. To obtain the probability distribution of a random variable, a lot of samples are needed to apply the statistics inference approach. Chapter 4 and 5 provide two uncertain DEA methods to evaluate the DMUs with limited or insufficient statistical data, named fuzzy DEA and uncertain DEA. In order to evaluate the DMUs in which uncertainty and randomness appear simultaneously, the hybrid DEA based on chance theory is presented in Chapter 6.

Operations Research is a field whose major contribution has been to propose a rigorous fonnulation of often ill-defmed problems pertaining to the organization or the design of large scale systems, such as resource allocation problems, scheduling and the like. While this effort did help a lot in understanding the nature of these problems, the mathematical models have proved only partially satisfactory due to the difficulty in gathering precise data, and in formulating objective functions that reflect the multi-faceted notion of optimal solution according to human experts. In this respect linear programming is a typical example of impressive achievement of Operations Research, that in its detenninistic fonn is not always adapted to real world decision-making : everything must be expressed in tenns of linear constraints ; yet the coefficients that appear in these constraints may not be so well-defined, either because their value depends upon other parameters (not accounted for in the model) or because they cannot be precisely assessed, and only qualitative estimates of these coefficients are available. Similarly the best solution to a linear programming problem may be more a matter of compromise between various criteria rather than just minimizing or maximizing a linear objective function. Lastly the constraints, expressed by equalities or inequalities between linear expressions, are often softer in reality that what their mathematical expression might let us believe, and infeasibility as detected by the linear programming techniques can often been coped with by making trade-offs with the real world.

Since its birth, the field of Probabilistic Logic Programming has seen a steady increase of activity, with many proposals for languages and algorithms for inference and learning. This book aims at providing an overview of the field with a special emphasis on languages under the Distribution Semantics, one of the most influential approaches. The book presents the main ideas for semantics, inference, and learning and highlights connections between the methods. Many examples of the book include a link to a page of the web application http://cplint.eu where the code can be run online. This 2nd edition aims at reporting the most exciting novelties in the field since the publication of the 1st edition. The semantics for hybrid programs with function symbols was placed on a sound footing. Probabilistic Answer Set Programming gained a lot of interest together with the studies on the complexity of inference. Algorithms for solving the MPE and MAP tasks are now available. Inference for hybrid programs has changed dramatically with the introduction of Weighted Model Integration. With respect to learning, the first approaches for neuro-symbolic integration have appeared together with algorithms for learning the structure for hybrid programs. Moreover, given the cost of learning PLPs, various works proposed language restrictions to speed up learning and improve its scaling.

Robust optimization is still a relatively new approach to optimization problems affected by uncertainty, but it has already proved so useful in real applications that it is difficult to tackle such problems today without considering this powerful methodology. Written by the principal developers of robust optimization, and describing the main achievements of a decade of research, this is the first book to provide a comprehensive and up-to-date account of the subject. Robust optimization is designed to meet some major challenges associated with uncertainty-affected optimization problems: to operate under lack of full information on the nature of uncertainty; to model the problem in a form that can be solved efficiently; and to provide guarantees about the performance of the solution. The book starts with a relatively simple treatment of uncertain linear programming, proceeding with a deep analysis of the interconnections between the construction of appropriate uncertainty sets and the classical chance constraints (probabilistic) approach. It then develops the robust optimization theory for uncertain conic quadratic and semidefinite optimization problems and dynamic (multistage) problems. The theory is supported by numerous examples and computational illustrations. An essential book for anyone working on optimization and decision making under uncertainty, Robust Optimization also makes an ideal graduate textbook on the subject.