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Special section: Methods for Uncertainty Computing Either Uncertainty Propagation or Optimization Under Uncertainty

# Optimization Based Algorithms for Uncertainty Propagation Through Functions With Multidimensional Output Within Evidence Theory

[+] Author and Article Information
Christian Gogu1

Université de Toulouse, INSA, UPS, Mines Albi, ISAE, ICA (Institut Clément Ader), 118, route de Narbonne, F-31062 Toulouse, Francechristian.gogu@gmail.com

Youchun Qiu

Université de Toulouse, Institut de Mathématiques de Toulouse, 118 route de Narbonne, F-31062 Toulouse, France

Stéphane Segonds, Christian Bes

Université de Toulouse, INSA, UPS, Mines Albi, ISAE, ICA (Institut Clément Ader), 118, route de Narbonne, F-31062 Toulouse, France

1

Corresponding author.

J. Mech. Des 134(10), 100914 (Sep 28, 2012) (8 pages) doi:10.1115/1.4007393 History: Received January 14, 2012; Revised June 11, 2012; Published September 21, 2012; Online September 28, 2012

## Abstract

Evidence theory is one of the approaches designed specifically for dealing with epistemic uncertainty. This type of uncertainty modeling is often useful at preliminary design stages where the uncertainty related to lack of knowledge is the highest. While multiple approaches for propagating epistemic uncertainty through one-dimensional functions have been proposed, propagation through functions having a multidimensional output that need to be considered at once received less attention. Such propagation is particularly important when the multiple function outputs are not independent, which frequently occurs in real world problems. The present paper proposes an approach for calculating belief and plausibility measures by uncertainty propagation through functions with multidimensional, nonindependent output by formulating the problem as one-dimensional optimization problems in spite of the multidimensionality of the output. A general formulation is first presented followed by two special cases where the multidimensional function is convex and where it is linear over each focal element. An analytical example first illustrates the importance of considering all the function outputs at once when these are not independent. Then, an application example to preliminary design of a propeller aircraft then illustrates the proposed algorithm for a convex function. An approximate solution found to be almost identical to the exact solution is also obtained for this problem by linearizing the previous convex function over each focal element.

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## Figures

Figure 1

Example of Dempster–Shafer subsets structure. The shaded area represents event e.

Figure 2

Belief and plausibility representation in evidence theory

Figure 3

BPA uncertainty structure assumed on the input parameters. Upper numbers provide the bounds on the variables while lower numbers provide the corresponding BPA.

Figure 4

Graphical representation of the uncertainty structure in the output space (yellow, magenta, green, and blue rectangles) together with the failure region (red dotted rectangle)

Figure 5

Example of matching chart for preliminary aircraft sizing, typical of a propeller airplane

Figure 6

BPA uncertainty structure assumed on the input parameters. Upper numbers provide the bounds on the variables while lower numbers provide the corresponding BPA.

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