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The Picture fuzzy linguistic set (PFLS) is an extension of the intuitionistic fuzzy set (IFS) and linguistic variables (LVs), which has been applied successfully in the process of decision making. Considering the lack of closeness of extant PFLS operations and the interrelationship among input attributes do not consider. In this paper, for the sake of addressing those limitations, we firstly redefine some novel operational laws for PFLS by introducing linguistic scale functions and the related properties are studied. Then, a novel score function and accuracy function are also defined to compare PFLSs. Subsequently, in consideration of the superiority of the Muirhead Mean (MM) operator in capturing the interaction relationship between the input parameters, we extend the MM operator to the Picture fuzzy linguistic context and then propose Picture fuzzy linguistic weighted MM operator and its dual form in a new light. After that, these operators have adopted to build two novel models to solve multiple attribute decision-making (MADM) problems. Finally, a practical example for the selection of the innovative "Mobike" sharing bike design is provided to illustrate the practicality and effectiveness of proposed approaches.

A novel picture fuzzy linguistic Muirhead Mean aggregation operators and their application to multiple attribute decision making