Dempster-Shafer Theory in the Analysis and Design of Uncertain Engineering Systems

Abstract

A methodology for the analysis and design of uncertain engineering systems in the presence of multiple sources of evidence based on Dempster-Shafer Theory (DST) is presented. DST can be used when it is not possible to obtain a precise estimation of system response due to the presence of multiple uncertain input parameters. The information for each of the uncertain parameters is assumed to be available in the form of interval-valued data from multiple sources implying the existence of large epistemic uncertainty in the parameters. The DST approach, in conjunction with the use of the vertex method, and the evidence-based fuzzy methodology are used in finding the response of the system. A new method, called Weighted Dempster Shafer Theory for Interval-valued data (WDSTI), is proposed for combining evidence when different credibilities are associated with different sources of evidence. The application of the methodology is illustrated by considering the safety analysis of a welded beam in the presence of multiple uncertain parameters. The epistemic uncertainty can be modeled using fuzzy set theory. In order to extend the mathematical laws of crisp numbers to fuzzy theory, we can use the extension principle, which provides a methodology that fuzzifies the parameters or arguments of a function, resulting in computable fuzzy sets. In this work, an uncertain parameter is modeled as a fuzzy variable and the available evidences on the ranges of the uncertain parameter, in the form of basic probability assignments (bpa’s), are represented in the form of membership functions of the fuzzy variable. This permits the use of interval analysis in the application of a fuzzy approach to uncertain engineering problems. The extension of DST in the decision making of uncertain engineering systems based on using different combination rules such as Yager’s rule, Inagaki’s extreme rule, Zhang’s center combination rule and Murphy’s average combination rule is also presented with an illustrative application.