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.

Theory and practice of uncertain programming

Reasonable properties for the ordering of fuzzy quantities (II)

This paper provides a survey of credibility theory that is a new branch of mathematics for studying the behavior of fuzzy phenomena. Some basic concepts and fundamental theorems are introduced, including credibility measure, fuzzy variable, membership function, credibility distribution, expected value, variance, critical value, entropy, distance, credibility subadditivity theorem, credibility extension theorem, credibility semicontinuity law, product credibility theorem, and credibility inversion theorem. Recent developments and applications of credibility theory are summarized. A new idea on chance space and hybrid variable is also documented.

A survey of credibility theory

This chapter is an overview of past and present works dealing with fuzzy intervals and their operations. A fuzzy interval is a fuzzy set in the real line whose level-cuts are intervals. Particular cases include usual real numbers and intervals. Usual operations on the real line canonically extend to operations between fuzzy quantities, thus extending the usual interval (or error) analysis to membership functions. What is obtained is a counterpart of random variable calculus, but where, contrary to the latter case, there is no compensation between variables. Many results pertaining to basic properties of fuzzy interval analysis are summed up in the chapter. Computational methods are presented, exact or approximate ones, based on parametric representations, or level-cut approximations. The generalized fuzzy variable calculus involving interactive variables is also discussed with emphasis on triangular-norm based fuzzy additions. Dual ‘optimistic’ operations on fuzzy intervals, i.e., with maximal error compensation are also presented; its interest lies in providing tools for solving fuzzy interval equations. This chapter also contains a reasoned survey of methods for comparing and ranking fuzzy intervals. The chapter includes some historical background, as well as pointers to applications in mathematics and engineering.

González Muñoz

Papers

A subjective approach for ranking fuzzy numbers