Weldon A. Lodwick

Interval analysis and fuzzy set theory

An overview of interval analysis, its development, and its relationship to fuzzy set theory is given. Possible areas of further fruitful research are highlighted.

Fuzzy linear programming using a penalty method

In this paper we begin with a standard form of the linear programming problem. We replace each constant in the problem with a fuzzy number. We then reformat the objective and constraints into an unconstrained fuzzy function by penalizing the objective for possible constraint violations. The range of this fuzzy function lies in the space of fuzzy numbers. The objective is then redefined as optimizing the expected midpoint of the image of this fuzzy function. We show that this objective defines a concave function which, therefore, can be maximized globally. We present an algorithm for finding the optimum.

Improving the Efficiency of Stochastic Dominance Techniques Using Convex Set Stochastic Dominance

The advantages of convex set stochastic dominance (CSD) are discussed in terms of extending other stochastic dominance criteria in a way which will decrease Type II errors (large efficient sets) without increasing the Type I errors (inaccurate rankings). An empirical example ranking pest management strategies demonstrates the potential of CSD by reducing the efficiency set by almost 60% without imposing additional constraints on the preference set. It is suggested that CSD may permit more imprecise representations of risk preferences, avoiding utility measurement problems, and still identify efficient sets of acceptable sizes.

Solving large-scale fuzzy and possibilistic optimization problems

Fuzzy and possibilistic optimization methods are demonstrated to be effective tools in solving large-scale problems. In particular, an optimization problem in radiation therapy with various orders of complexity from 1000 to 62,250 constraints for fuzzy and possibilistic linear and nonlinear programming implementations possessing (1) fuzzy or soft inequalities, (2) fuzzy right-hand side values, and (3) possibilistic right-hand side is used to demonstrate that fuzzy and possibilistic optimization methods are tractable and useful. We focus on the uncertainty in the right side of constraints which arises, in the context of the radiation therapy problem, from the fact that minimal and maximal radiation tolerances are ranges of values, with preferences within the range whose values are based on research results, empirical findings, and expert knowledge, rather than fixed real numbers. The results indicate that fuzzy/possibilistic optimization is a natural and effective way to model various types of optimization under uncertainty problems and that large fuzzy and possibilistic optimization problems can be solved efficiently.

Interval and Fuzzy Analysis: A Unified Approach

Publisher Summary A unified approach to real-valued interval and fuzzy analysis emphasizing common themes is presented in this chapter. Interval analysis and fuzzy analysis are viewed as a bridge between deterministic problem solving and problems with generalized uncertainty. The chapter discusses the basic knowledge of interval analysis and fuzzy set theory. Interval analysis and fuzzy set theory, as active fields of research and application, are relatively new mathematical disciplines receiving the impetus that defined them as separate fields. The connection between interval analysis and possibility theory arose out of fuzzy set theory. The theory of interval analysis models, among other things, the uncertainty arising from numerical computation, which can be considered a source of ambiguity. Fuzzy set theory and possibility theory model, among other things, the uncertainty of vagueness and ambiguity arising from the transitional nature of entities and a lack of information. The chapter also discusses the distinction between fuzzy set theory and possibility theory.

Analysis of structure in fuzzy linear programs

Abstract Fuzzy linear programming problems are analyzed within the fuzzy set context to uncover redundancies, infeasibilities, variables whose values are fixed, and implied bounds on rows and columns. Applications include pre-analysis of perturbed constraint matrices and post-optimization analysis fuzzy linear programming problems.

Constructing consistent fuzzy surfaces from fuzzy data

Given fuzzy data describing a three-dimensional entity such as terrain, we develop methods to construct surfaces that are consistent with the uncertainty in the data and surface model itself. In particular, surfaces generated from higher α-cut values of the fuzzy data are contained within the surfaces generated by lower α-cut values of the fuzzy data. Moreover, the smoothness and continuity conditions of the surface generating method is maintained by each level surface. We demonstrate the ideas by developing two- and three-dimensional surfaces from fuzzy data for cubic splines and digital terrain models generated from triangulation.

Estimating and Validating the Cumulative Distribution of a Function of Random Variables: Toward the Development of Distribution Arithmetic

A method for estimating and validating the cumulative distribution of a function of random variables (independent or dependent) is presented and examined. The method creates a sequence of bounds that will converge to the distribution function in the limit for functions of independent random variables or of random variables of known dependencies. Moreover, an approximation is constructed from and contained in these bounds. Preliminary numerical experiments indicate that this approximation is close to the actual distribution after a few iterations. Several examples are given to illustrate the method.

Optimality conditions of type KKT for optimization problem with interval-valued objective function via generalized derivative

This paper addresses the optimization problems with interval-valued objective function. For this we consider two types of order relation on the interval space. For each order relation, we obtain KKT conditions using of the concept of generalized Hukuhara derivative (