Due to the uncertainty and complexity of socioeconomic environments and cognitive diversity of decision makers, the cognitive information over alternatives provided by decision makers is usually uncertain and fuzzy. Two-tuple linguistic Pythagorean fuzzy sets (2TLPFSs) provide useful tools to depict the uncertain and fuzzy cognitions of the decision makers over attributes. To effectively handle such common cases, in this paper, some power Muirhead mean (PMM) operator and power dual MM (PDMM) operator operators under 2TLPFS environment are proposed and investigated the methods for multiple attribute decision making(MADM) problems based on the PMM and PDMM operators with 2-tuple linguistic Pythagorean fuzzy numbers (2TLPFNs) are investigated. Firstly, some new PMM and PDMM operators to aggregate 2-tuple linguistic Pythagorean fuzzy cognitive information is developed, such as 2-tuple linguistic Pythagorean fuzzy MM (2TLPFPMM) operator, 2-tuple linguistic Pythagorean fuzzy weighted PMM (2TLPFWPMM) operator, 2-tuple linguistic Pythagorean fuzzy PDMM (2TLPFPDMM) operator, and 2-tuple linguistic Pythagorean fuzzy weighted PDMM (2TLPFNWPDMM) operator, which consider the interrelationship of 2TLPFNs, and can generate more accurate results than the existing aggregation operators. After that, the developed aggregation operator are applied to MADM with 2TLPFNs and two MADM methods are designed, which can be applied to different decision making situations. Based on the proposed operators and built models, two methods are developed to solve the MADM problems with 2TLPFNs and the validity and advantages of the proposed method are analyzed by comparison with some existing approaches. The method proposed in this paper can effectively handle the MADM problems in which the attribute information is expressed by 2TLPFNs, the attributes’ weights are completely known, and the attributes are interactive. Finally, an example for green supplier selection is used to show the proposed methods.

由于社会经济环境的不确定性和复杂性以及决策者的认知多样性，决策者提供的关于替代方案的认知信息通常是不确定和模糊的。两元语言的勾股勾股模糊集（2TLPFS）提供了有用的工具，用于描述决策者对属性的不确定性和模糊性认知。为了有效地应对这种常见情况，本文提出了一些2TLPFS环境下的幂Muirhead均值（PMM）算子和幂对偶MM（PDMM）算子，并研究了基于PMM的多属性决策（MADM）问题的方法。研究了具有二元语言毕达哥拉斯模糊数（2TLPFN）的PDMM算子。首先，开发了一些新的PMM和PDMM运算符以聚合2元组语言的勾股模糊认知信息，例如2元组语言的勾股模糊MM（2TLPFPMM）运算符，2元语言的勾股模糊加权PMM（2TLPFWPMM）运算符，2元语言毕达哥拉斯模糊PDMM（2TLPFPDMM）运算符和2元组语言毕达哥拉斯模糊加权PDMM（2TLPFNWPDMM）运算符，它们考虑了2TLPFN的相互关系，并且可以产生比现有聚合运算符更准确的结果。然后，将开发的聚合算子应用于具有2TLPFN的MADM，设计了两种MADM方法，可以分别应用于不同的决策情况。根据拟议的运营商和构建的模型，提出了两种解决2TLPFN的MADM问题的方法，并与现有方法进行了比较，分析了该方法的有效性和优势。本文提出的方法可以有效地解决MADM问题，即用2TLPFN表示属性信息，完全知道属性的权重，并且属性是交互式的。最后，以绿色供应商选择为例来说明所提出的方法。