The Linguistic generalzied Fuzzy Set (FS) is more efficient and effective for depicting awkward and uncertain data compared to existing models. In this manuscript, we describe the Sugeno-Weber laws for linguistic generalzied fuzzy information. Because these operational laws will help us in the construction of the “power aggregation operators” for linguistic Linear Diophantine Fuzzy Sets (LDFSs), called “Power Averaging (PA) operator”, “Power Weighted Averaging (PWA)”, “Power Geometric (PG)”, and “Power Weighted Geometric (PWG)” for linguistic linear Diophantine fuzzy values. These models can help us aggregate the collection of data into a singleton set very easily. Additionally, we investigate the model of the multi-attributive border approximation area comparison technique for derived operators to enhance the effectiveness of the proposed theory. The problem of supply management is used for the integration of environmentally friendly procedures into supply chain management techniques, where the model of sustainable sourcing, eco-design, waste management, energy efficiency, transportation, and collaboration are the major parts of the considered theory. For this, we illustrate some numerical problems for evaluating the problem of supply chain theory by using the proposed models. Finally, we deliberate on the power and strength of the suggested models by comparing the value of the proposed and existing models.