## 1 Introduction  
Various tailings storage facility (TSF) failures have occasionally occurred around the world in recent years, such as the 2008 980 Gou TSF failure in China [1], the 2015 Feijao TSF failure and the 2019 Fundao TSF failure in Brazil [2], the 2014 Mount Polley TSF failure in Canada [3], the 1985 Prestaval fluorite mine tailings dam failure in Stava, Italy [3], and so on. And huge loss of lives and properties have been caused by the occurrences of TSF failure [4]. To avoid and reduce the unfavorable effects caused by TSF failure, it is significantly important for decision makers (DMs) or managers to select a desirable scheme to reduce the risk of TSF failure. Decision-making problems in the risk reduction of TSFs are complex and challenging, due to multiple attributes and their inaccurate variables, and the uncertainty of decision information. Therefore, decision-making problems in risk reduction scheme (RRS) selection are multi-attribute decision-making problems, and the research and application of decision-making methods for risk reduction is crucial. Multiple-attribute decision-making problems refer to the selection of a favorable alternative associated with multiple attributes. In some previous studies [5, 6, 7], DMs are usually considered to be completely rational individuals in the decision process, in which the psychological behavior of DMs is overlooked. For example, the technique for order preference by similarity to the ideal solution (TOPSIS), based on the utility theory, usually has a poor performance in distinguishing whether the alternative meets the criteria in detail [8, 9]. The preference ranking organization method for enrichment of evaluations (PROMETHEE) fails to consider psychological factors [10]. The elimination and choice translating reality (ELECTRE) is an inefficient method due to its excessive parameters and complex processes [11]. Indeed, different DMs have diverse attitudes toward unknown and uncertain risks; therefore, taking their risk appetites into account is important in the selection of RRSs. The classical TODIM (an acronym for interactive and multiple-attribute decision-making in Portuguese) model is a multicriteria method proposed by [PERSON] and [PERSON] [12, 13]. It is a useful method derived from a prospective function for capturing psychological behavior under risk. The decision-making results within the TODIM model are aimed at maximizing the value function, which is determined by the calculation of the dominance of one alternative over another. Consequently, the TODIM method can be regarded as an effective reflector of the behavioral characteristics and risk appetites of DMs [14]. It has been researched and broadly applied to determine the ranking order of alternatives in various decision-making aspects, such as site selection [15], risk evaluation [16], and failure mode identification [16, 17]. For example, based on the cloud and TODIM method, [PERSON] et al. [18] proposed a new method to evaluate emergency plans. The new method could provide a reference for the improvement of emergency management capabilities. [PERSON] et al. [19] presented an extended TODIM model of risk evaluation of excavation systems, which was adopted in an excavation construction project and provided a comprehensive risk status perception of excavation. [PERSON] et al. [20] developed a modified TODIM method to study the security of the rare earth industry in China, which introduced a new evaluation of rare earth elements' security. [PERSON] and [PERSON] [21] proposed a risk decision analysis method based on the TODIM method, where the psychological behavior of DMs was considered. The TODIM method was improved by [PERSON] et al. [17], to determine the risk priority of failure modes in the failure mode and effect analysis problem. In the actual decision-making problems, linguistic attribute values could reflect the DMs' preference information more directly compared with other expression forms. The selection of linguistic variables is an important problem to be solved.