A picture fuzzy multi-attribute decision-making approach based on Hamacher Muirhead mean operators

Picture fuzzy set (PFS) concepts are modified versions of fuzzy sets. Picture fuzzy sets cover all aspects of portraying human opinion accurately. In this paper, we create Muirhead mean (MM) operators employing arithmetic operations modelled by Hamacher t-norm (TN) and t-conorm (TCN) using picture fuzzy information. These operators are known as picture fuzzy. Hamacher MM (PFHMM) and picture fuzzy Hamacher weighted MM (PFHWMM). Combining Hamacher t-norm and t-conorm arithmetic with the MM operator allows for flexible aggregation and consideration of attribute interrelationships. Also, MM is a generalization of commonly used aggregation operators, including arithmetic mean (AM), geometric mean (GM), Bonferroni mean (BM), and Maclaurin symmetric mean (MSM). The paper discusses some desirable properties and exceptional cases of proposed operators. The study also examines the Multiple Attribute Decision Making (MADM) technique using the suggested PFHWMM operator under the system of PFS information. A MADM problem is about allocating healthcare resources during a pandemic to test how well the suggested operators and methods work. To demonstrate the superiority of the currently proposed methods, we conducted a comprehensive comparative analysis to contrast the results of these approaches with the prevailing theories in the literature.