Abstract: Multiple attribute decision making method is one of the current hot research subjects and has been applied to a large amount of practical decision problems. With the development of society, owing to the lack of information or decision makers' hesitation, traditional crisp numbers are difficult to deal with this situation. Different types of uncertain numbers are proposed, such as fuzzy number, intuitionistic fuzzy number, linguistic term, Pythagorean fuzzy number. When decision makers evaluate some subjective information, they may feel comfortable to use linguistic terms. However, when they have some hesitation on their opinions with linguistic terms, they may use Pythagorean fuzzy uncertain linguistic numbers. Therefore, solving the multiple attribute decision making problems with Pythagorean fuzzy uncertain linguistic numbers is meaningful. In some cases, attribute weights may be unknown and attributes are interactional. To cope with this issue, a multiple attribute decision making method is proposed based on grey incidence analysis and Heronian mean (HM) operator. Firstly, the Pythagorean fuzzy uncertain linguistic Heronian mean (PFULHM) operator and the Pythagorean fuzzy uncertain linguistic geometric Heronian mean (PFULGHM) operator are proposed and their properties including idempotence, monotonicity, boundedness and commutativity are proved. Considering the attribute weights with different importance, based on the above two operators, the Pythagorean fuzzy uncertain linguistic weighted Heronian mean (PFULWHM) operator and the Pythagorean fuzzy uncertain linguistic weighted geometric Heronian mean (PFULWGHM) operator are defined. Furthermore, a novel grey incidence analysis formula for Pythagorean fuzzy uncertain linguistic numbers, and a new method are proposed to obtain attribute weights using grey incidence analysis. Finally, a decision model based on PFULWHM operator and PFULWGHM operator is constructed and an example is provided to illustrate the effectiveness of the proposed method. By means of the case study, we can draw the advantages of our method: (1)avoiding the loss of information; (2)solving decision making problem effectively with interactional attributes.

Key words: multiple attribute decision making, Pythagorean fuzzy uncertain linguistic set, Pythagorean fuzzy uncertain linguistic weighted Heronian mean operator, grey incidence analysis