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# 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.

### Keywords

Dempster-Shafer Theory Engineering design Evidence Fuzzy### References

- Agarwal, H., Renaud, J. E., Preston, E. L., and Padmanabhan, D. (2004), “Uncertainty quantification using evidence theory in multidisciplinary design optimization”,
*Reliability Engineering and System Safety*, Vol. 85, pp. 281–294.CrossRefGoogle Scholar - Alim, S. (1988), “Application of Dempster-Shafer theory for interpretation of seismic parameters”,
*Journal of Structural Engineering*, Vol. 114, No. 9, pp. 2070–2084.CrossRefGoogle Scholar - Bae, H. R., Grandhi, R. V. and Canfield, R. A. (2004), “Epistemic uncertainty quantification techniques including evidence theory for large scale structures”,
*Computers and structures*, Vol. 80, pp. 1101–1112.CrossRefGoogle Scholar - Beynon, M., Curry, B. and Morgan, P. (2000), “The Dempster-Shafer theory of evidence: An alternative approach to multicriteria decision modeling”,
*Omega*. Vol. 28, pp. 37–50.CrossRefGoogle Scholar - Butler, A. C., Sadeghi, F., Rao, S. S. and LeClair, S. R. (1995), “Computer-aided design/ engineering of bearing systems using Dempster-Shafer theory”,
*Artificial Intelligence for Engineering Design, Analysis and Manufacturing*, Vol. 9, pp. 1–11.CrossRefGoogle Scholar - Dong, W. M., and Shah, H. C. (1987), “Vertex method for computing functions of fuzzy variables”,
*Fuzzy Sets and Systems*, Vol. 24, pp. 65–78.MATHMathSciNetCrossRefGoogle Scholar - Ferson, S., Kreinovich, V. Ginzburg, L., Myers, D. S. and Sentz, K. (2003). “Constructing probability boxes and Dempster-Shafer Structures”,
*SAND Report*.Google Scholar - Ha-Rok, B. (2004), “Uncertainty quantification and optimization of structural response using evidence theory”, PhD Dissertation, Wright State University.Google Scholar
- Inagaki, T. (1991), “Interdependence between Safety-Control Policy and Multiple-Sensor Schemes Via Dempster-Shafer Theory”, IEEE Transactions on Reliability 40(2):182–188.MATHMathSciNetCrossRefGoogle Scholar
- Klir, G. J. and M. J. Wierman (1998),
*Uncertainty-Based Information: Elements of Generalized Information Theory*, Physica-Verlag, Heidelberg.MATHGoogle Scholar - Kozine, I. O. and Utkin, L. V. (2004), “An Approach to Combining Unreliable Pieces of Evidence and their Propagation in a System Response Analysis,”
*Reliability Engineering and System Safety*, Vol. 85, No. 1–3, pp. 103–112.CrossRefGoogle Scholar - Murphy, C. K. (2000), “Combining belief functions when evidence conflicts”,
*Decision Support Systems*29, pp. 1–9.CrossRefGoogle Scholar - Oberkampf, W. L., and Helton J. C., “Investigation of Evidence Theory for Engineering Applications,”
*Non-Deterministic Approaches Forum*, Denver, CO, April-2002, Paper No. AIAA-2002–1569.Google Scholar - Parikh, C. R., Pont, M. J. and Jones, N. B. (2001), “Application of Dempster- Shafer Theory in condition monitoring systems: A case study”,
*Pattern Recognition Letters*, Vol. 22, No. 6–7, pp. 777–785.MATHCrossRefGoogle Scholar - Rao, S. S., and Sawyer, J. P., “A fuzzy element approach for the analysis of imprecisely-defined systems,” AIAA Journal, Vol. 33, No. 12, 1995, pp. 2364–2370.MATHCrossRefGoogle Scholar
- Rao, S. S. and Annamdas, K. K. (2008), “Evidence-Based Fuzzy Approach for the Safety Analysis of Uncertain Systems”,
*AIAA Journal*, Vol. 46, No. 9, pp. 2383–2387.CrossRefGoogle Scholar - Sentz, K. (2002), “Combination of evidence in Dempster-Shafer theory”, PhD Dissertation, Binghamton University.CrossRefGoogle Scholar
- Shafer, G. (1976),
*A Mathematical Theory of Evidence*, Princeton University Press, Princeton, NJ.MATHGoogle Scholar - Tanaka, K. and G. J. Klir (1999), “A design condition for incorporating human judgement into monitoring systems”,
*Reliability Engineering and System Safety*, Vol. 65, No. 3, pp. 251–258.CrossRefGoogle Scholar - Yager, R. (1987a), “On the Dempster-Shafer framework and new combination rules”,
*Information Sciences*41, pp. 93–137.MATHMathSciNetCrossRefGoogle Scholar - Zadeh, L., “Fuzzy sets,”
*IEEE Information Control*, Vol. IC-8, 1965, pp. 338–353.MathSciNetCrossRefGoogle Scholar - Zhang, L. (1994), “Representation, independence, and combination of evidence in the Dempster-Shafer theory”,
*Advances in the Dempster-Shafer Theory of Evidence*(Eds. R. R. Yager, J. Kacprzyk and M. Fedrizzi), John Wiley, New York, pp. 51–69.Google Scholar