**1. Introduction**

Multi-attribute decision-making (MADM) aims to select the best alternative solution(s) from multiple alternatives and has been widely used in various fields [1–3]. In MADM problems, linguistic terms are a convenient and natural way to describe evaluation information. For example, the decision-makers (DMs) can use linguistic terms such as 'Extremely low', 'Very low', 'Low', 'Fair', 'High', 'Very high', and 'Extremely high' to estimate service quality, product performance, and so forth. Therefore, MADM problems based on linguistic terms have received increasing attention. In [4], Aggarwal proposed a new aggregation operator for linguistic terms, and the effectiveness of the operator was illustrated by a case study on the supplier selection problem. Jin [5] developed two group decision-making methods to handle MADM problems under linguistic set environment, and comparative analysis with other methods was performed to demonstrate the validity and merits of the two methods. Yu [6] proposed an extended TODIM method with unbalanced hesitant fuzzy linguistic term sets for MADM problems. For linguistic decision-making problems, Pei [7] developed a new decision-making method by integrating the fuzzy linguistic multiset and TOPSIS methods, and two practical examples were utilized to verify the feasibility of the proposed approach. For the venture capital problem under a linguistic environment, Cheng [8] proposed an interaction approach. However, the methods mentioned above directly replace the linguistic variables with linguistic subscript in the decision-making process, which may cause distortion of information. To better express linguistic variables, the linguistic 2-tuple [9,10] and linguistic scale function [11,12] were introduced to deal with linguistic sets. Nevertheless, the linguistic 2-tuple and linguistic scale function methods still use linguistic subscript to express language variables in nature. Besides, it is difficult to explain the rationality in theory by simply replacing linguistic word with its linguistic subscript. Furthermore, people may have diverse opinions on identical words, but linguistic subscript can only depict a single meaning for one person, which may lead to information distortion.

As distinct from the linguistic 2-tuple and scale function, shadowed sets [13,14] can effectively construct linguistic terms using a data-driven method, and have recently been attracting more and more attention [15,16]. The membership value of the shadowed set is not a precise number, and its distribution is composed of three different zones: the core zone, shadowed zone, and exclusion zone. The core zone and the exclusion zone take the values of 1 and 0, which means all the elements of both zones are fully compatible with or completely excluded from the linguistic word described by a shadowed set. The shadowed zone is an entire unit interval perceived as a zone of uncertainty, which means we are not sure whether the shadowed zone elements represent the linguistic word described by a shadowed set.

In addition, in many situations, experts may hesitate as to what attribute values should be given by them, due to the increasing complexity. Consequently, Atanassov [17] proposed the intuitionistic fuzzy set (IFS) to express uncertainty, which involves not only membership degree but also non-membership degree. However, the limitation of IFS is that the sum of membership degree and non-membership degree must be no more than 1, which makes it difficult to sufficiently express the ideas of the DMs. Therefore, Yager [18] defined the Pythagorean fuzzy set (PFS), which can effectively express the certainty and uncertainty of experts. Recently, PFS has been introduced to deal with MADM problems [19–22]. Zhang and Xu [19] proposed the operation rules of PFS, and extended the TOPSIS method to PFS. By combining PFS with the hesitant fuzzy set (HFS), a new fuzzy set was defined by Liang and Xu [20], named the hesitant Pythagorean fuzzy set (HPFS), an extended TOPSIS method with HPFS was subsequently proposed. Zhang [21] extended PFS to the interval-valued case, and explored the basic operation rules of the Pythagorean fuzzy set (IVPFS). In addition, a Pythagorean fuzzy QUALIFLEX method was developed by integrating closeness index, and its effectiveness was demonstrated through a hierarchical MADM problem. Combining PFSs with linguistic variables, the definition of Pythagorean fuzzy linguistic set (PFLS) was proposed by Peng and Yang [22] and the operation rules of PFLS was defined, subsequently.

Inspired by the idea of the shadowed set and PFS, we propose a new approach to solve MADM problems under linguistic set environment. Firstly, we define Pythagorean shadowed set and explore some theorems of the shadowed set. Secondly, a score function of the Pythagorean shadowed number is defined and the detailed decision-making procedures-based upon the score function is proposed. Finally, a case study of supplier selection is adopted to verify the feasibility of the proposed approach.

The organization of this paper is as follows. Section 2 presents the preliminaries of the Pythagorean fuzzy set and shadowed set. In Section 3, the shadowed set model of seven-level language term is obtained by a data-driven method. A new score function of Pythagorean shadowed number is introduced in Section 4. Section 5 mainly addresses a new MADM method based on Pythagorean shadowed set. The effectiveness of the proposed approach is demonstrated through a supplier selection problem in Section 6, and comparative analysis is made with the other existing methods. Finally, some conclusions are drawn in Section 7.
