Interval linear systems as a necessary step in fuzzy linear systems

International audienceThis article clarifies what it means to solve a system of fuzzy linear equations, relying on the fact that they are a direct extension of interval linear systems of equations, already studied in a specific interval mathematics literature. We highlight four distinct definitions of a systems of linear equations where coefficients are replaced by intervals, each of which based on a generalization of scalar equality to intervals. Each of the four extensions of interval linear systems has a corresponding solution set whose calculation can be carried out by a general unified method based on a relatively new concept of constraint intervals. We also consider the smallest multidimensional intervals containing the solution sets. We propose several extensions of the interval setting to systems of linear equations where coefficients are fuzzy intervals. This unified setting clarifies many of the anomalous or inconsistent published results in various fuzzy interval linear systems studies

Two numerical methods for the solutions of optimal control problems with computed error bounds using the maximum principle of pontryagin

A simplification of the proof of the maximum principle of Pontryagin is obtained for constrained and unconstrained optimal control problems. Two numerical methods for solving optimal control problems with guaranteed error bounds using the maximum principle of Pontryagin and interval analysis are derived

Constrained Interval Arithmetic

: This paper presents an approach to solving the long-standing dependency problem in interval arithmetic. An extension to interval arithmetic, called here constrained interval arithmetic, is developed. Unlike interval arithmetic, constrained interval arithmetic has an additive inverse, a multiplicative inverse and satisfies the distributive law. This means that the algebraic structure of constrained interval arithmetic is different than that of interval arithmetic. The applicability of constrained interval arithmetic is explored. 1. Introduction: It is well-known in the interval analysis literature that interval arithmetic overestimates the resultant width of the interval when dependencies are present. This overestimation can be arbitrarily large (see, for example, (Neumaier 1990, pages 16-19)). Example 1Consider y = f(x) = x(1 \Gamma x); x 2 [0; 1]: The implementation of interval arithmetic yields y = [0; 1] \Theta (1 \Gamma [0; 1]) = [0; 1] \Theta [0; 1] = [0; 1]. The actual range..

Application of Remote Sensing and Geographic Information System Techniques to Evaluate Agricultural Production Potential in Developing Countries

The Comprehensive Resource Inventory and Evaluation System (CRIES) project uses a multidisciplinary approach to (1) assist developing countries to analyze their agricultural production potential, and (2) enhance their capabilities to conduct such analyses for policy formulation and evaluation. The project staff has collaborated within their respective disciplines with counterparts in the Dominican Republic, Costa Rica, Nicaragua, Syria, and Honduras to enhance each country\u27s capacity to analyze agricultural production possibilities. The technical assistance provided has been in one of four functional areas. These are: (1) data acquisition; (2) information management and analyses; (3) analyses of policy alternatives; and (4) technology transfer and training. The paper emphasizes the use of remote sensing in data acquisition and the decision analyses followed in deciding the appropriateness of alternative geographic information systems for use in developing countries

A new look at nonlinear dynamics in solids with inelastic neutron scattering spectroscopy

Gradual numbers have been introduced recently as a means of extending standard interval computation methods to fuzzy intervals. The literature treats monotonic functions of fuzzy intervals. In this paper, we combine the concepts of gradual numbers and optimization, which allows for the evaluation of any differentiable function on fuzzy intervals, with no monotonicity requirement.

Flexible and generalized uncertainty optimization: theory and methods

This book presents the theory and methods of flexible and generalized uncertainty optimization. Particularly, it describes the theory of generalized uncertainty in the context of optimization modeling. The book starts with an overview of flexible and generalized uncertainty optimization. It covers uncertainties that are both associated with lack of information and that more general than stochastic theory, where well-defined distributions are assumed. Starting from families of distributions that are enclosed by upper and lower functions, the book presents construction methods for obtaining flexible and generalized uncertainty input data that can be used in a flexible and generalized uncertainty optimization model. It then describes the development of such a model in detail. All in all, the book provides the readers with the necessary background to understand flexible and generalized uncertainty optimization and develop their own optimization model.