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The arithmetics of two dimensional belief functions

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Abstract

Real-world information is imperfect and usually characterized by uncertainty and partial reliability. Given these limitations, Z-numbers is introduced as a more suitable concept for describing real-world information. In recent years, Z-numbers has received immense attention. However, the general approach for computations over Z-number is too complex. Thus a simpler mathematical model has been proposed—Two Dimensional Belief Function (TDBF). A TDBF is an ordered pair of belief functions; the first belief function is on the frame of discernment of the values that a variable can take, and the second belief function is a measure of reliability of the first belief function. However, the processing of TDBF-based information requires a new theory to be developed, together with new approaches and procedures for computation with TDBF. In this paper, we propose a general framework for computations over TDBF, comprising addition, subtraction, multiplication, division, square and square root of TDBFs. In particular, a general approach to implementing combination over two TDBFs is also given. Our proposed method is validated by a variety of numerical examples, with an application in decision making problem.

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Acknowledgments

The authors greatly appreciate the reviews’ suggestions and the editor’s encouragement. The work is partially supported by National Natural Science Foundation of China (Grant No. 61973332), JSPS Invitational Fellowships for Research in Japan (Short-term).

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Authors and Affiliations

  1. Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu, 610054, China

    Yangxue Li & Yong Deng

  2. Faculty of Communications Sciences, University of Teramo, Teramo, Italy

    Danilo Pelusi

  3. Science, Mathematics & Technology Cluster, Singapore University of Technology and Design, Singapore, Singapore

    Kang Hao Cheong

  4. School of Education, Shannxi Normal University, Xi’an, 710062, China

    Yong Deng

  5. School of Knowledge Science, Japan Advanced Institute of Science and Technology, Nomi, Ishikawa, 923-1211, Japan

    Yong Deng

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  1. Yangxue Li
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  2. Danilo Pelusi
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  4. Yong Deng
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Correspondence to Kang Hao Cheong or Yong Deng.

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Li, Y., Pelusi, D., Cheong, K.H. et al. The arithmetics of two dimensional belief functions. Appl Intell 52, 4192–4210 (2022). https://doi.org/10.1007/s10489-021-02435-x

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