The induced generalized interval-valued intuitionistic fuzzy hybrid Shapley averaging operator and its application in decision making
Introduction
Aggregation operators, as an important research topic in decision-making theory, have been researched by many scholars [3], [4], [5], [10], [13], [15], [18], [20], [30], [31], [33], [44], [45], [46], [47], [48], [49], [52], [53], [54]. One of the most importance aggregation operators is the ordered weighted averaging (OWA) operator [45], whose fundamental aspect is a reordering step in which the input arguments are rearranged in descending order and the weight vector is merely associated with its ordered position. Since it was first proposed in 1988, many extending forms are developed, such as the continuous OWA (COWA) operator [48], the generalized OWA (GOWA) operator [49], the generalized intuitionistic fuzzy OWA (GIFOWA) operator [54] the induced continuous OWA (ICOWA) operator [5], the induced generalized OWA (IGOWA) operator [13], the induced generalized continuous OWA (IGCOWA) operator [5], the generalized intuitionistic fuzzy OWA (GIFOWA) operator [10], the quasi-arithmetic intuitionistic fuzzy OWA (QAIFOWA) operator [50] and the induced generalized intuitionistic fuzzy OWA (IG-IFOWA) operator [20]. Later, some authors found the OWA operator only considers the importance of the elements’ ordered positions, but does not give the importance of their own [31]. In 2003, Xu and Da [31] proposed the hybrid weighted averaging (HWA) operator, which not only considers the ordered position of the argument, but also gives the importance of the argument. After the pioneering work of Xu and Da [31], many developed forms are proposed, such as the hybrid weighted arithmetical averaging (HWAA) operator [11], the hybrid weighted geometric mean (HWGM) operator [32], the continuous hybrid weighted quasi-arithmetical averaging (C-HWQA) operator [11], the generalized intuitionistic fuzzy hybrid aggregation (GIFHA) operator [56], the induced generalized hybrid averaging (IGHA) operator [12], the intuitionistic fuzzy Einstein hybrid aggregation (IFEHA) operator [55], the linguistic hybrid geometric mean (LHGA) operator [34] and the uncertain linguistic hybrid aggregation (ULHA) operator [35].
All above hybrid aggregation operators are based on the assumption that the discussed elements in a set are independent, i.e., they only consider the addition of the importance of individual elements. However, in many practical situations, the elements in a set are usually correlative [7], [8], [16], [22], [23], [24], [25], [26], [27], [36], [37], [47], [53], [56]; for example, Grabisch [7] gave the following classical example: “We are to evaluate a set of students in relation to three subjects: {mathematics, physics, literature}, we want to give more importance to science-related subjects than to literature, but on the other hand we want to give some advantage to students that are good both in literature and in any of the science-related subjects”. Therefore, we need to find some new ways to deal with these situations in which the decision data in a question are correlative. Fuzzy measures [21], as an affective tool to measure the interaction among elements, have been successfully used in decision making [7], [8], [16], [17], [22], [23], [24], [25], [26], [27], [36], [37], [47], [53], [56]. At present, we have not found the research about the hybrid aggregation operators based on fuzzy measures. Furthermore, how to obtain the fuzzy measure on attribute set is not considered. In order to deal with these issues, we here develop a new interval-valued intuitionistic fuzzy hybrid operator, which is named as the induced generalized interval-valued intuitionistic fuzzy hybrid Shapley averaging (IG-IVIFHSA) operator. Moreover, when the information about the weights of attributes and their ordered positions is incompletely known, the models for the optimal fuzzy measures on them are established, respectively. Based on the given operator and the models for the optimal fuzzy measures, an approach to multi-attribute decision making under interval-valued intuitionistic fuzzy environment is developed.
The rest parts of this paper are organized as follows. In Section 2, some basic concepts and notations are reviewed, such as the HWA, HWAA and IGHA operators, fuzzy measures, the Shapley function, interval-valued intuitionistic fuzzy sets (IVIFSs). In Section 3, the IG-IVIFHSA operator is defined. Some important cases and desirable properties are studied. In Section 4, we first give the models for obtaining the optimal fuzzy measures on attribute set and ordered set. Then, an approach to interval-valued intuitionistic fuzzy multi-attribute decision making with incomplete weight information and interactive conditions is developed. In Section 5, two examples are provided to illustrate the developed procedure. The conclusion is made in the last section.
Section snippets
Preliminaries
For the convenience of analysis, some basic concepts are reviewed to facilitate future discussions.
A new interval-valued intuitionistic fuzzy hybrid aggregation operator
Based on above analysis, in this section we shall define the induced generalized interval-valued intuitionistic fuzzy hybrid Shapley averaging (IG-IVIFHSA) operator, and consider some important cases. Meantime, some properties are studied.
An approach to interval-valued intuitionistic fuzzy multi-attribute decision making
This section presents a new approach to interval-valued intuitionistic fuzzy multi-attribute decision making, which does not only consider the importance of attributes and their ordered positions, but also reflect the correlations among attributes and among their ordered positions. When the weights of attributes and their ordered positions are given, we can apply the introduced aggregation operator to develop a method to interval-valued intuitionistic fuzzy multi-attribute decision making.
Two practical examples
In this section, we use two decision making problems that were discussed in [29], [51] to demonstrate how to apply the proposed approach. Example 1 Suppose there is an investment company, which wants to invest a sum of money in the best option (adapted from Ref. [51]). There is a panel with four possible alternatives to invest the money: (1) a1 is a car company; (2) a2 is a food company; (3) a3 is a computer company; and (4) a4 is an arms company. The investment company must take a decision according to
Conclusion
At present, most research about hybrid aggregation operators is based on the assumption that the elements in a set are independent. As some scholars pointed, this assumption is unreasonable in many situations, where there exists some degree of inter-dependent or correlative characteristics. In order to deal with this issue, we propose the induced generalized interval-valued intuitionistic fuzzy hybrid Shapley averaging (IG-IVIFHSA) operator, which does not only globally consider the importance
Acknowledgments
The authors first gratefully thank the Editor-in-Chief Hamido Fujita and two anonymous referees for their valuable comments, which have much improved the paper. This work was supported by the National Natural Science Foundation Youth Project of China (No. 71201089), the National Natural Science Foundation of China (Nos. 71071018 and 71271217) and the Natural Science Foundation Youth Project of Shandong Province, China (ZR2012GQ005).
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