A novel decision model with Einstein aggregation approach for garbage disposal plant site selection under $q$-rung orthopair hesitant fuzzy rough information

Environmental science and pollution research has benefits around the globe. Human activity produces more garbage throughout the day as the world's population and lifestyles rise. Choosing a garbage disposal site (GDS) is crucial to effective disposal. In illuminated of the advancements in society, decision-makers concede a significant challenge for assessing an appropriate location for a garbage disposal site. This research used a multi-attribute decision-making (MADM) approach based on $q$-rung orthopair hesitant fuzzy rough ( $q$-ROHFR) Einstein aggregation information for evaluating GDS selection schemes and providing decision-making (DM) support to select a suitable waste disposal site. In this study, first, q-ROHFR Einstein average aggregation operators are integrated. Some intriguing characteristics of the suggested operators, such as monotonicity, idempotence and boundedness were also explored. Then, a MADM technique was established using the novel concept of $q$-ROHFR aggregation operators under Einstein t-norm and t-conorm. In order to help the decision makers (DMs) make a final choice, this technique aims to rank and choose an alternative from a collection of feasible alternatives, as well as to propose a solution based on the ranking of alternatives for a problem with conflicting criteria. The model's adaptability and validity are then demonstrated by an analysis and solution of a numerical issue involving garbage disposal plant site selection. We performed a the sensitivity analysis of the proposed aggregation operators to determine the outcomes of the decision-making procedure. To highlight the potential of our new method, we performed a comparison study using the novel extended TOPSIS and VIKOR schemes based on $q$-ROHFR information. Furthermore, we compared the results with those existing in the literature. The findings demonstrate that this methodology has a larger range of information representation, more flexibility in the assessment environment, and improved consistency in evaluation results.