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

House of Quality

Nonlinear Programming: A Unified Approach

Are Investors Reluctant to Realize Their Losses

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

Uncertainty theory as a branch of axiomatic mathematics has been widely used to deal with human uncertainty. The two commonly used numerical characteristics of uncertain variables, the expected value and the variance together with their mathematical properties have been discussed and applied to real optimization problems in an uncertain environment. As a further study, in this paper, we focus on the covariance and correlation coefficient of uncertain variables. The definitions and calculation formulae of covariance and correlation coefficient of two uncertain variables are suggested by means of their inverse distributions. Then we show that the correlation coefficient of uncertain variables is essentially a measure of the relevance of distributions of uncertain variables. Finally, the relation between variance and covariance is analysed and represented with some equalities and inequalities.

The covariance of uncertain variables: definition and calculation formulae

In this paper, the minimum spanning tree problem is investigated on the graph with fuzzy edge weights. The notion of fuzzy α-minimum spanning tree is presented based on the credibility measure, and then the solutions of the fuzzy α-minimum spanning tree problem are discussed under different assumptions. First, we respectively, assume that all the edge weights are triangular fuzzy numbers and trapezoidal fuzzy numbers and prove that the fuzzy α-minimum spanning tree problem can be transformed to a classical problem on a crisp graph in these two cases, which can be solved by classical algorithms such as the Kruskal algorithm and the Prim algorithm in polynomial time. Subsequently, as for the case that the edge weights are general fuzzy numbers, a fuzzy simulation-based genetic algorithm using Prufer number representation is designed for solving the fuzzy α-minimum spanning tree problem. Some numerical examples are also provided for illustrating the effectiveness of the proposed solutions.

Fuzzy α-minimum spanning tree problem: definition and solutions

A novel two-phase approach for the bi-objective simultaneous delivery and pickup problem with fuzzy pickup demands

Bayesian networks have been widely applied to domains such as medical diagnosis, fault analysis, and preventative maintenance. In some applications, because of insufficient data and the complexity ...

Fuzzy belief propagation in constrained Bayesian networks with application to maintenance decisions

In this paper, an analysis of the convergence performance is conducted for a class of possibilistic clustering algorithms (PCAs) utilizing the Zangwill convergence theorem. It is shown that under certain conditions the iterative sequence generated by a PCA converges, at least along a subsequence, to either a local minimizer or a saddle point of the objective function of the algorithm. The convergence performance of more general PCAs is also discussed.

Jian Zhou

Papers

The covariance of uncertain variables: definition and calculation formulae

Fuzzy α-minimum spanning tree problem: definition and solutions

A novel two-phase approach for the bi-objective simultaneous delivery and pickup problem with fuzzy pickup demands

Fuzzy belief propagation in constrained Bayesian networks with application to maintenance decisions

On the convergence of some possibilistic clustering algorithms