## 1 Introduction
The IFS could better describe uncertain information through the concepts of membership, non-membership, and hesitation degrees. It is therefore more appropriate to address situations in which the membership value of the element in the set may be hesitant or uncertain. Given the advantages of the IFS, it has been combined with several risk assessment techniques [18]. [PERSON] [19] applied the IFS to FMEA for risk analysis. In the IFS theory, the language variables represented by intuitionistic fuzzy numbers were used to evaluate and transform risk factors. [PERSON] [20] presented a hybrid approach of intuitional fuzzy and TOPSIS for risk matrix analysis. Subsequently, [PERSON] et al. [21] used IF-AHP to address uncertain data in risk-based maintenance investment optimization. [PERSON] et al. [22] provided an IFS-based MCDM sorting model for improving HAZOP analysis. [PERSON] and [PERSON] [23] proposed a novel system for safety and risk assessment. This method combines the concept of BN with FTA under an intuitionistic fuzzy environment. Whether the FS or IFS, explicit membership and/or non-membership functions are required. As commonly known, in IF-BT models, occurrence possibilities of basic events are represented by the IFS obtained through expert elicitation. However, it is often troublesome to require experts to express the membership and non-membership degrees with crisp values, since the membership and non-membership functions may not be precisely known in many cases. For this reason, the IFS is extended to interval-valued intuitionistic fuzzy sets (IVIFSs). In IVIFSs, the membership and non-membership degrees are represented by interval values in the range of [0, 1] rather than crisp values in the range of [0, 1], which bring interval freedom to the representation and treatment of uncertainty, Therefore, the IVIFS method has become increasingly popular. [PERSON] et al. [24] incorporated an IVIFS into a hybrid MCDM model of risk analysis to handle vagueness and uncertainty. [PERSON] et al. [25] employed an IVIFS to express the assessment values of risk factors in FMEA. [PERSON] [26] combined an IVIFS and rough number theory to propose an IVIF rough number concept and its corresponding FMEA model. [PERSON] and [PERSON] [27] also proposed an IVIF cloud theory based on FMEA for risk evaluation. The aforementioned studies have shown that the IVIFS approach is a valuable solution to handle uncertainty in risk and safety assessment. However, to the best of our knowledge, no methods exist for evaluating SB effectiveness involving the IVIFS concept. Therefore, this paper proposes an IVIF-BT method for safety assessment and SB evaluation by introducing an IVIFS into the BT model. In this method, the failure possibilities of basic events with/without SBs in the BT model are represented as interval-valued intuitionistic fuzzy numbers (IVIFNs) to characterize the hesitation of events. Also, an IVIF-AHP is employed to analyze the importance weights of different severity types of failure consequences. Furthermore, the SB effectiveness is clearly defined and measured by comparing system failure risk with SBs to that without SBs. In this approach, the BT model effectively links key events with the associated causes and consequences while fully and comprehensively considering the role of SBs within the model. Compared to other methods, this approach not only allows for safety assessments but also considers the failure risks and consequences of the system in the presence of various SBs, further analyzing the effectiveness of each SB. Additionally, in this method, obtaining key data through expert elicitation enhances the applicability of the model, and the use of IVIFNs helps handle the uncertainty and fuzziness of expert opinions, thereby increasing the reliability of the model. Consequently, this method can effectively address the challenges of multiple uncertainties and incomplete information in process industries, ensuring the accuracy and credibility of safety assessments.