[Paper Review] D numbers theory: a generalization of Dempster-Shafer evidence theory

Yong Deng|arXiv (Cornell University)|May 13, 2014

Multi-Criteria Decision Making34 references5 citations

TL;DR

This paper introduces D numbers theory (DNT) as a generalization of Dempster-Shafer evidence theory that relaxes the strict mutual exclusivity requirement in the frame of discernment. By introducing an exclusive coefficient and a new combination rule, DNT enables flexible fusion of uncertain, non-exclusive linguistic evidence, with numerical results showing successful application in linguistic decision-making scenarios.

ABSTRACT

Efficient modeling of uncertain information in real world is still an open issue. Dempster-Shafer evidence theory is one of the most commonly used methods. However, the Dempster-Shafer evidence theory has the assumption that the hypothesis in the framework of discernment is exclusive of each other. This condition can be violated in real applications, especially in linguistic decision making since the linguistic variables are not exclusive of each others essentially. In this paper, a new theory, called as D numbers theory (DNT), is systematically developed to address this issue. The combination rule of two D numbers is presented. An coefficient is defined to measure the exclusive degree among the hypotheses in the framework of discernment. The combination rule of two D numbers is presented. If the exclusive coefficient is one which means that the hypothesis in the framework of discernment is exclusive of each other totally, the D combination is degenerated as the classical Dempster combination rule. Finally, a linguistic variables transformation of D numbers is presented to make a decision. A numerical example on linguistic evidential decision making is used to illustrate the efficiency of the proposed D numbers theory.

Motivation & Objective

• To address the limitation in Dempster-Shafer theory that requires mutually exclusive hypotheses in the frame of discernment, which often fails in real-world applications involving linguistic variables.
• To develop a systematic framework for D numbers theory (DNT) that allows non-exclusive, incomplete, and imprecise information to be modeled and combined effectively.
• To introduce a discounting mechanism based on an exclusive coefficient to adjust D numbers before fusion, improving reliability in conflicting or overlapping evidence.
• To propose a linguistic variables transformation method to convert fused D numbers into final decision outputs for practical use in decision-making contexts.
• To demonstrate the effectiveness of DNT in handling linguistic evidential decision-making problems where classical evidence theory fails due to non-exclusivity.

Proposed method

• Proposes D numbers as a generalization of basic probability assignments in Dempster-Shafer theory, allowing non-exclusive hypotheses in the frame of discernment.
• Introduces an exclusive coefficient to measure the degree of non-exclusivity among hypotheses, with a value of 1 indicating full exclusivity.
• Develops a new combination rule for D numbers that incorporates the exclusive coefficient to discount evidence before fusion, ensuring consistency and reducing overconfidence.
• Applies a transformation process to convert the final fused D number into linguistic variables (e.g., Very Poor, Fair, Good) for interpretable decision outcomes.
• Uses a numerical example with three experts assessing criteria using linguistic terms, each represented as triangular fuzzy numbers, to demonstrate the fusion process.
• Employs a discounting mechanism where D numbers are adjusted based on the exclusive coefficient before applying the combination rule to ensure robustness in non-exclusive settings.

Experimental results

Research questions

• RQ1 Can D numbers theory effectively model uncertain information when hypotheses in the frame of discernment are not mutually exclusive, as in linguistic decision-making?
• RQ2 How can a reliable combination rule be designed for D numbers that accounts for the degree of non-exclusivity among hypotheses?
• RQ3 What is the impact of the exclusive coefficient on the fusion outcome, and how does it improve the reliability of evidence combination?
• RQ4 Can D numbers be transformed into linguistic variables to support interpretable final decisions in real-world applications?
• RQ5 Under what conditions does D numbers theory reduce to classical Dempster-Shafer evidence theory?

Key findings

• The proposed D numbers theory successfully handles non-exclusive linguistic evidence, overcoming a key limitation of classical Dempster-Shafer theory.
• The exclusive coefficient enables adaptive discounting of D numbers, with higher values indicating greater exclusivity and reducing the influence of conflicting or overlapping evidence.
• When the exclusive coefficient is 1, the D number combination rule degenerates into the classical Dempster combination rule, confirming consistency with existing theory.
• In the numerical example, the final fused D number for criterion C1 showed the highest mass on {MG} (0.9351), indicating a strong expert consensus on 'Medium Good' rating.
• After linguistic transformation, the final decision for A1 was dominated by {MG} with a mass of 0.9351, followed by {F} (0.0504), confirming a clear preference for 'Medium Good' in the decision outcome.
• The method effectively handles conflicting and imprecise expert opinions, producing a coherent and interpretable final decision despite non-exclusive linguistic inputs.