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Canonical Decomposition of Basic Belief Assignment for Decision-Making Support

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Modelling and Development of Intelligent Systems (MDIS 2020)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1341))

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Abstract

We present a new methodology for decision-making support based on belief functions thanks to a new theoretical canonical decomposition of dichotomous basic belief assignments (BBAs) that has been developed recently. This decomposition based on proportional conflict redistribution rule no 5 (PCR5) always exists and is unique. This new PCR5-based decomposition method circumvents the exponential complexity of the direct fusion of BBAs with PCR5 rule and it allows to fuse quickly many sources of evidences. The method we propose in this paper provides both a decision and an estimation of the quality of the decision made, which is appealing for decision-making support systems.

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Notes

  1. 1.

    I.e. the solution, or the decision to take.

  2. 2.

    This is so-called Shafer’s model of FoD [5].

  3. 3.

    The complete ignorance is denoted \(\varTheta \) in Shafer’s book [1].

  4. 4.

    \(Disj(m_p,m_c)\) denotes symbolically the disjunctive fusion of \(m_p\) with \(m_c\).

  5. 5.

    Taking \(y=1\) would means that \(x(1-y)=0\) but \(m(A)=a\) with \(a\ne 0\) in general, so the choice of \(y=1\) is not possible.

  6. 6.

    The third equality \(m(A\cup \bar{A})=1-a-b=\frac{(1-x)(1-y)}{1-xy}\) being redundant with (10) and (11) is useless.

  7. 7.

    Otherwise the denominator of (10) and (11) will equal zero.

  8. 8.

    Because there is no simple analytical expressions for solutions x and y of PCR5-based canonical decomposition.

  9. 9.

    The solutions can be easily obtained with the roots command of Matlabâ„¢.

  10. 10.

    This point is not detailed here because is out of the scope of this paper.

  11. 11.

    Where the complexity is linear with the number of dichotomous BBAs to fuse.

  12. 12.

    For clarity, we need to introduce in the notations a superscript to indicate the FoD we are working on.

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Author information

Authors and Affiliations

  1. The French Aerospace Lab, Palaiseau, France

    Jean Dezert

  2. Department of Mathematics, University of New Mexico, Gallup, NM, USA

    Florentin Smarandache

Authors
  1. Jean Dezert
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  2. Florentin Smarandache
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Correspondence to Florentin Smarandache .

Editor information

Editors and Affiliations

  1. Lucian Blaga University of Sibiu, Sibiu, Romania

    Dana Simian

  2. Lucian Blaga University of Sibiu, Sibiu, Romania

    Laura Florentina Stoica

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Dezert, J., Smarandache, F. (2021). Canonical Decomposition of Basic Belief Assignment for Decision-Making Support. In: Simian, D., Stoica, L.F. (eds) Modelling and Development of Intelligent Systems. MDIS 2020. Communications in Computer and Information Science, vol 1341. Springer, Cham. https://doi.org/10.1007/978-3-030-68527-0_7

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  • DOI: https://doi.org/10.1007/978-3-030-68527-0_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-68526-3

  • Online ISBN: 978-3-030-68527-0

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