| Summary: | Hamacher aggregation operators are more flexible and more dominant to determine the interrelationships between any number of attributes. The goal of this manuscript is to elaborate on the principle of linear Diophantine uncertain linguistic sets and explored their useful Hamacher operational laws. The existing notions of intuitionistic uncertain linguistic sets, Pythagorean uncertain linguistic sets, and q-rung orthopair uncertain linguistic sets have certain applications in different fields. Unfortunately, these theories have their limitations related to the truth and falsity grades. To eradicate these limitations, the theory of linear Diophantine uncertain linguistic sets with the addition of reference parameters is massive flexible than the existing drawbacks. This notion removes the restrictions of prevailing methodologies, and the decision-maker can freely choose the grades without any limitations. This structure also categorizes the problem by changing the physical sense of reference parameters. Moreover, by using the investigated linear Diophantine uncertain linguistic information and Hamacher aggregation operators, we explored the linear Diophantine uncertain linguistic generalized Hamacher averaging operator and linear Diophantine uncertain linguistic generalized Hamacher hybrid averaging operator. Additionally, a multi-attribute decision-making (MADM) procedure is buildup based on the investigated operators under the linear Diophantine uncertain linguistic information. Certain numerical examples are illustrated by using initiated operators to determine the dominance and flexibility of explored operators. To find the consistency and supremacy of the presented operators, we compare the proposed work with certain prevailing operators and discussed their geometrical expressions to show that the introduced operators in this manuscript are extensively powerful and more useful than the prevailing drawbacks.
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