International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems

This paper presents the independence of fuzzy variables as well as its applications in fuzzy random optimization. First, the independence of fuzzy variables is defined based on the concept of marginal possibility distribution function, and a discussion about the relationship between the independent fuzzy variables and the noninteractive (unrelated) fuzzy variables is included. Second, we discuss some properties of the independent fuzzy variables, and establish the necessary and sufficient conditions for the independent fuzzy variables. Third, we propose the independence of fuzzy events, and deal with its fundamental properties. Finally, we apply the properties of the independent fuzzy variables to a class of fuzzy random programming problems to study their convexity.

A random fuzzy variable is a function from a credibility space to the set of random variables. Chance distribution is a type of mathematical description of random fuzzy variables. This paper presents a sufficient and necessary condition for chance distributions of random fuzzy variables.

Comparison of fuzzy variables is considered one of the most important topics in fuzzy theory. A new approach for ranking fuzzy variable via credibility measure — fuzzy dominance is presented in this paper. Some basic properties of fuzzy dominance are investigated. As an illustration, the cases of fuzzy dominance rule for triangular fuzzy variables are examined.

The concept of fuzzy entropy is used to provide a quantitative measure of the uncertainty associated with every fuzzy variable. This paper proposes the maximum entropy principle for fuzzy variables, that is, out of all the membership functions satisfying given constraints, choose the one that has maximum entropy. The problem is what is the specific formulation of the maximum entropy membership function. The purpose of this paper is to solve this problem by Euler–Lagrange equation.

A fundamental problem in fuzzy analysis is that the membership function is specified subjectively. In other words, the modelers specify the membership function form and assign the values of the parameters in membership function via their working experiences. Different from its probabilistic counterpart, fuzzy mathematical theory does not provide convenient parameter estimation approach. In this paper, we first review Liu's non-classical credibility measure theory (i.e., (∨,·)-credibility measure theory) in Liu⁷, because the fuzzy theory initiated by Zadeh¹⁰ contains a critical weakness: non self-duality property for possibility measure. We establish a parameter estimation of the membership function in terms of maximum entropy principle on the ground of self-dual credibility measure theory. Furthermore, based on the data assimilated membership function, we can calculate membership grades on the fuzzy environmental index, using PM¹⁰ air pollution as an illustration. We treat the calculated membership grades as spatially distributed random quantity, and therefore perform the standard ordinary kriging approach for generating the predicted environmental index map, for PM¹⁰ prediction map.

We present an application of the Analytic Hierarchy Process (AHP) in the sensitivity analysis of a multi-parameter decision support problem. In this problem, the Multi-Parametric Sensitivity Analysis (MPSA) appears appropriate and is employed. A drawback of MPSA is that it only produces a sensitivity comparison between every two parameters, and does not offer a systematic method for ranking all parameters at once. Without the overall sensitivity ranking, it is difficult to interpret the analysis result from MPSA, especially when the number of parameters is large. Our paper shows that AHP can be used to perform such ranking based on MPSA's pair-wise comparisons.

This paper presents a fuzzy programming method to design supply chain network, in which the customer demands and transportation costs are assumed to be fuzzy parameters. Existing researches on supply chain network design problem are either restricted on deterministic environment or only address stochastic parameters. In this paper, we consider this problem in fuzzy environment. Under different criteria, we format three types of models for the decision makers: expected cost optimization model, chance-constrained model and chance maximization model. A genetic algorithm based on fuzzy simulation is developed to solve the proposed fuzzy models. Moreover, some numerical examples are presented to illustrate the effectiveness of models and solution algorithm.

Real observations of qualities (continuous or discrete types) are not precise numbers but more or less non-precise. The best description of such data is by so-called fuzzy numbers. The fuzziness is different from measurement errors and stochastic uncertainty. In this paper we propose definitions of fuzzy mode, fuzzy median and fuzzy mean as well as investigation of their related properties and employ these techniques in the practical applications of real life. Empirical result shows that fuzzy statistics with soft computing is more realistic and reasonable for the statistical research. Finally certain comments are suggested for the further studies. We hope that this reformation will make the corresponding fuzzy techniques more appropriate and efficient to researchers whose only experience with statistics is using the traditional methods.

The degree-constrained minimum spanning tree problem (dc-MST) is to find a minimum spanning tree of the given graph, subject to constraints on node degrees. This paper investigates the dc-MST problem with fuzzy random weights. Three concepts are presented: expected fuzzy random dc-MST, (α,β)-dc-MST and the most chance dc-MST according to different optimization requirement. Correspondingly, by using the concepts as decision criteria, three fuzzy random programming models for dc-MST are given. Finally, a hybrid intelligent algorithm is designed to solve these models, and some numerical examples are provided to illustrate its effectiveness.

Fuzzy clustering is an approach using the fuzzy set theory as a tool for data grouping, which has advantages over traditional clustering in many applications. Many fuzzy clustering algorithms have been developed in the literature including fuzzy c-means and possibilistic clustering algorithms, which are all objective-function based methods. Different from the existing fuzzy clustering approaches, in this paper, a general approach of fuzzy clustering is initiated from a new point of view, in which the memberships are estimated directly according to the data information using the fuzzy set theory, and the cluster centers are updated via a performance index. This new method is then used to develop a generalized approach of possibilistic clustering to obtain an infinite family of generalized possibilistic clustering algorithms. We also point out that the existing possibilistic clustering algorithms are members of this family. Following that, some specific possibilistic clustering algorithms in the new family are demonstrated by real data experiments, and the results show that these new proposed algorithms are efficient for clustering and easy for computer implementation.

The paper deals with stochastic mathematical programming problems with additional difficulty brought up by the presence of infinitely many constraints. A sample average approximation (SAA) approach is proposed and properties of SAA estimators, analyzed. It is shown that for the case of deterministic constraints, SAA estimators are consistent and meet requirements to qualify the estimates as acceptable approximations of unknown values. Moreover, good error bounds of the estimate of the objective function are provided. We also discuss a Galerkin-like scheme and a tailored cutting-plane method for solving resulting semi-infinite SAA problems. The framework is then extended to the case of random constraints through Karhunen–Loève expansion or constraints approximation via sampling. A general procedure for solving stochastic semi-infinite programming problems, that lends itself better for parallel computation, is then described. Numerical examples are also included for the sake of illustration.