The Design, Implementation and Application of Z-number Linguistic Model

Abstract

Decision-making in the real world is often complicated by the uncertainty of information and its partial reliability. Not only might the exact value of a quantity be vague, but our confidence in that value can also be limited. Traditional approaches, such as probability theory and fuzzy set theory, each have their own limitations when handling the partial reliability of information. To overcome these limitations, Zadeh introduced Z-numbers, defined as ordered pairs Z = ( ˜ A, ˜B ), integrating a fuzzy quantity with its associated reliability measure. This thesis builds on the concept of Z-numbers to develop a comprehensive Z-number linguistic modeling framework, explicitly addressing the uncertainty in both value and reliability. The main contributions include: (1) Z-number linguistic model: A formal model is established for representing linguistic information (such as expert statements or sensor observations) as Z-number linguistic terms. This structured representation simultaneously encodes fuzzy values and their reliability, laying theoretical groundwork for robust reasoning under uncertainty. (2) Z-number-valued rule-based classification system (ZRBCS): Extending traditional fuzzy rulebased classification systems, ZRBCS incorporates Z-number conditions into rules. Unlike conventional fuzzy systems, each rule condition explicitly encodes uncertainty and reliability, enabling nuanced inference and improving classification performance by utilizing negative samples to determine reliability and adjust fuzzy partitions. (3) Z-number-valued rule-based decision tree (ZRDT): Building on fuzzy rule-based decision trees, ZRDT integrates Z-numbers into the decision-tree framework. Information gain replaces fuzzy confidence for feature selection, and negative samples guide the refinement of fuzzy numbers for better data fitting. Experimental comparisons demonstrate that ZRDT achieves higher classification accuracy and produces a more compact decision tree than standard algorithms such as FRDT, PUBLIC, C4.5, and AdaBoost.NC. (4) Z-number generation model: A novel nonlinear model, Maximum Expected Minimum Entropy (MEME), is proposed to generate Z-numbers directly from multiple probability distributions, eliminating reliance on expert input. Further, MEME is integrated into a Z-valuation rule-based (ZVRB) classification system, significantly enhancing decision-making performance under uncertainty. Experimental validation confirms that the ZVRB system outperforms classical classifiers and existing rule-based methods. (5) Optimization via FURIA: The Fuzzy Unordered Rule Induction Algorithm (FURIA) is integrated as an optimization tool to refine the proposed Z-number models. FURIA improves accuracy and interpretability by dynamically adjusting membership functions, optimizing rule sets, and selecting predictive features. This demonstrates that the developed framework can adaptively learn from data, significantly enhancing overall performance. Collectively, these contributions advance uncertainty modeling by explicitly managing both value uncertainty and information reliability, providing a robust and nuanced toolset for real-world decision-making beyond existing probabilistic and fuzzy frameworks.