# A Three-Way Acceleration Approach for Interval-Valued Multi-Attribute Decision-Making Problems
## 1 Introduction
### Related Work  
The author in [4] proposes a fuzzy set, which has become an important tool for describing fuzzy information and dealing with uncertain problems. Authors in [5] apply neutrosophic similarity measure theory and the distance function to a decision-making model on COVID-19 pandemic patients. Authors in [6] propose a Variable Neighborhood Search (VNS) algorithm based on Multi-Attribute Decision-Making (MADM) strategy under a possibility environment. Under single valued pentapartitioned neutrosophic sets (SVPNS) environment, authors in [7] present a new similarity measure, namely tangent similarity measure, and validate the proposed SVPNS-MADM model. To select an appropriate tender for assigning the work, authors in [8] develop the multi-criteria decision-making model using a single-valued neutrosophic set. In addition, the basic definitions and properties of the Interval-Valued Intuitionistic Fuzzy Sets (IVIFSs) are presented in [9]. Under the setting of IVIFSs, the authors in [10] propose an optimization model to determine the attribute weights of the MADM problems. The truth degree of an IVIFS is represented by some closed sub-interval of [0, 1] rather than a single value in [0, 1]. Furthermore, the authors in [11] propose the concept of a complex fuzzy set, aiming at the case that the co-domain of a fuzzy set is a set of complex numbers instead of a sub-interval of [0, 1]. Unlike fuzzy numbers, an interval number does not strictly limit the membership degree within [0, 1]. Instead, it is represented by interval, making the meaning more intuitive and accurate. The research on interval numbers can be roughly summarized into two categories, comparison of interval numbers and applications of interval numbers. Interval number comparison means comparing sizes by ordering, and interval-number applications are mainly used in decision-making. The previous studies usually separate these two types of problems, and this paper fills this gap. We not only propose an interval number comparison method but also solve the interval-valued multi-attribute decision-making problem. As a granular computing method, three-way decision (TWD) theory [12; 13; 14] deals with many uncertain problems. The author in [15] presents a TWD model employing decision-theoretic rough sets (DTRS). With the help of the Bayesian theory, TWD [16; 17] is a methodology that can divide a universal set into three parts with the help of the Bayesian formula. The decision rules are as follows. Positive regions (POS) generate certain rules to decide acceptance, negative regions (NEG) generate certain rules to decide rejection, and boundary regions (BND) decide non-commitment or deferent. The positive and negative regions make strategies for certain objects, and the boundary region makes strategies for uncertain objects [18]. MADM makes a final decision based on many factors that satisfy various demands. Whether it is the MADM or the TWD theory, both provide reasonable methods to choose objects through appropriate criteria [19; 20]. Therefore, as the decision-making environment becomes increasingly complex, both are more closely related in decision analysis. The authors in [21] introduced TWD to fuzzy multi-attribute decision-making for the first time and suggested a method to establish the loss function. To better represent the membership and non-membership of the attributes, the authors in [22] construct the TWD model under the intuitionistic fuzzy number. The new model of TWD is proposed in [23], and the decision-making procedure is gained by analyzing Pythagorean fuzzy information systems. The authors in [24] present a novel TWD method to solve the MADM problems in fuzzy information systems. The authors in [25] propose a multi-attribute decision-making method based on the TWD theory in a hesitant fuzzy environment. Based on the above research, exploring the combination of the TWD theory and interval-valued multi-attribute decision-making is possible.