Contents I. Introduction
Since the theory of fuzzy set(FS) was proposed by Zadeh [1] in 1965, because of its effective description about the vagueness and imprecise of information, it has been attracting much attention of researchers all over the world. Atanassov [2] extends the theory of fuzzy set to the intuitionistic fuzzy set(IFS), which is characterized by a membership function, a non-membership function and a hesitancy function. It is proved that the IFS can describe the imprecise and uncertainty decision-making information more suitable. Atanassov and Gargov [3] propose the concept of the interval-valued intuitionistic fuzzy set(IVIFS) which is the extensive form of IFS. The IVIFS is characterized by the membership function and non-membership function with intervals rather than the crisp numbers. Multiple attribute decision-making (MADM) can be characterized as a process of choosing the best one from a set of alternatives with respect to some attributes or ranking the order of the alternatives. Its theory and methods are widely applied to various domains, such as economy, administration and military. A large amount of methods have been introduced to tackle the MADM problems under interval-valued intuitionistic fuzzy environment [4]–[10]. Xu [11] utilizes the Choquet integral to develop some intuitionistic fuzzy aggregation operators. The operators not only consider the importance of the elements or their ordered positions, but also can reflect the correlations among the elements or their ordered positions. Ye [12] proposes a fuzzy cross-entropy of the interval-valued intuitionistic fuzzy set to derive the optimal evaluation for the weight of each alternative. Yu et al. [13] proposes the interval-valued intuitionistic fuzzy prioritized weighted average (IVIFPWA) operator, the interval-valued intuitionistic fuzzy prioritized weighted geometric (IVIFPWG) operator to capture the prioritization phenomenon among the aggregated arguments. Yue [14] develops an approach for aggregating attribute satisfactory interval and attribute dissatisfactory interval into the collective attribute interval-valued intuitionistic fuzzy number. Li [15] develops a methodology for solving MADM problems with both ratings of alternatives on attributes and weights being expressed with IVIF sets by constructing a pair of nonlinear fractional programming models. Lakshmana [16] introduces and studies a new method for ranking interval-valued intuitionistic fuzzy sets. Wang et al. [17] propose an approach to multiple attribute decision making with incomplete attribute weight information under interval-valued intuitionistic fuzzy set environment. There are few researches about how to tackle this type of MADM where the decision-making information and attribute weight vector are both given by IVIFS. Zhang and Yu [18] presents an optimization model to determine attribute weights for MADM problems with incomplete weight information of criteria under IVIFS environment. A series of mathematical programming models based on cross-entropy are constructed and eventually transformed into a single mathematical programming model to determine the weights of attributes. Tan [19] develops an extension of TOPSIS to investigate the group decision-making problem in interval-valued intuitionistic fuzzy environment where inter-dependent or interactive characteristics among criteria and preference of decision makers are taken into account. Wang and Liu [20] introduce some Einstein geometric operators on interval-valued intuitionistic fuzzy sets, such as Einstein product, Einstein exponentiation etc., to investigate thedecision-making problem where individual assessments are provided as IVIFN. Ye [21] proposes an extended technique for order preference by similarity to ideal solution (TOPSIS) method for group decision making with interval-valued intuitionistic fuzzy numbers to solve the partner selection problem under incomplete and uncertain information environment. Chen et al. [22] propose the interval-valued intuitionistic fuzzy weighted average operator based on the traditional weighted average method and the Karnik-Mendel algorithms. Then, a fuzzy ranking method for intuitionistic fuzzy values based on likelihood-based comparison relations between intervals is proposed. Park et al. [23] extend the TOPSIS method to solve multiple attribute group decision making (MAGDM) problems in interval-valued intuitionistic fuzzy environment. We try to propose one feasible method to deal with it according to the former researches' results. A series of mathematical goal programming s are given to aggregate the decision-making information of each alternative into an interval-valued intuitionistic fuzzy number. Then we apply the TOPSIS method to rank all the alternatives according to their corresponding comprehensive values.
